The difference between length, height, and distance is fundamental in mathematics and real-life problem solving. These three terms are often used interchangeably in casual conversation, but in geometry, physics, and trigonometry, each has a specific meaning. Let’s break them down clearly. Length Length is the measurement of something from end to end. It typically refers to the longest side of an object or the measurement along a straight line. Example: The length of a pencil is 15 cm. In mathematics, when we say “length of a line segment,” we mean the numerical value that tells us how far apart its endpoints are. Length can be horizontal, vertical, or diagonal depending on the orientation. Height Height specifically refers to the vertical measurement from the bottom (base) to the top. It is always measured perpendicular to the ground or a reference base. Example: The height of a building is 50 meters. Height becomes important in trigonometry when we calculate the angle of elevation or depression. Whenever you see a problem involving towers, trees, or poles, the “height” is the vertical side of the right triangle. Distance Distance refers to the separation between two points in space. It may be along a straight line (shortest path) or along a curved path. Example: The distance between two towns is 25 km. In trigonometry, distance usually refers to the horizontal distance between an observer and the object being viewed. It is the base of the right triangle formed with height and the line of sight. Practical Example Imagine you are standing 30 meters away from a flagpole. The height is the vertical measurement of the flagpole, say 20 meters. The distance between you and the base of the flagpole is 30 meters. The length could refer to the size of the flag itself, say 5 meters from one end to another. Thus, in one scenario, all three terms can appear but have very different meanings. Why does this matter for students? Students preparing for exams often confuse these terms. Misunderstanding them can lead to incorrect identification of triangle sides in trigonometric problems. To avoid errors: Always identify height as vertical. Always identify distance as the horizontal ground separation. Use length in a general sense for measuring objects or lines.

The difference between length, height, and distance is fundamental in mathematics and real-life problem solving. These three terms are often used interchangeably in casual conversation, but in geometry, physics, and trigonometry, each has a specific meaning. Let’s break them down clearly. Length Length is the measurement of something from end to end. It typically refers to the longest side of an object or the measurement along a straight line. Example: The length of a pencil is 15 cm. In mathematics, when we say “length of a line segment,” we mean the numerical value that tells us how far apart its endpoints are. Length can be horizontal, vertical, or diagonal depending on the orientation. Height Height specifically refers to the vertical measurement from the bottom (base) to the top. It is always measured perpendicular to the ground or a reference base. Example: The height of a building is 50 meters. Height becomes important in trigonometry when we calculate the angle of elevation or depression. Whenever you see a problem involving towers, trees, or poles, the “height” is the vertical side of the right triangle. Distance Distance refers to the separation between two points in space. It may be along a straight line (shortest path) or along a curved path. Example: The distance between two towns is 25 km. In trigonometry, distance usually refers to the horizontal distance between an observer and the object being viewed. It is the base of the right triangle formed with height and the line of sight. Practical Example Imagine you are standing 30 meters away from a flagpole. The height is the vertical measurement of the flagpole, say 20 meters. The distance between you and the base of the flagpole is 30 meters. The length could refer to the size of the flag itself, say 5 meters from one end to another. Thus, in one scenario, all three terms can appear but have very different meanings. Why does this matter for students? Students preparing for exams often confuse these terms. Misunderstanding them can lead to incorrect identification of triangle sides in trigonometric problems. To avoid errors: Always identify height as vertical. Always identify distance as the horizontal ground separation. Use length in a general sense for measuring objects or lines.

The terms distance and height are central in trigonometry and geometry, yet they often cause confusion. Let’s make a clear distinction. Height Defined as the vertical measurement from base to top. Always measured perpendicular to the ground or reference line. Example: The height of a streetlight is 10 meters. In trigonometry, height is the opposite side when calculating with sine, cosine, or tangent ratios. Distance Refers to the horizontal ground measurement between two points. Example: The distance from a person to the base of the streetlight is 15 meters. In trigonometry, distance usually represents the adjacent side of the right triangle. Trigonometric Context Suppose a student is asked: “A person is standing 15 m away from a streetlight of 10 m height. What is the angle of elevation of the top?” Height = 10 m (vertical side). Distance = 15 m (horizontal side). Formula used: tan(θ)= Distance/Height =10/15 This shows how both terms appear together in most height-and-distance problems. Everyday Relevance Architects measure the height of buildings but need the distance from roads to design sightlines. Pilots calculate the height of descent and distance to the runway simultaneously. Surveyors use angles to measure the height of hills from a certain distance.

The terms distance and height are central in trigonometry and geometry, yet they often cause confusion. Let’s make a clear distinction. Height Defined as the vertical measurement from base to top. Always measured perpendicular to the ground or reference line. Example: The height of a streetlight is 10 meters. In trigonometry, height is the opposite side when calculating with sine, cosine, or tangent ratios. Distance Refers to the horizontal ground measurement between two points. Example: The distance from a person to the base of the streetlight is 15 meters. In trigonometry, distance usually represents the adjacent side of the right triangle. Trigonometric Context Suppose a student is asked: “A person is standing 15 m away from a streetlight of 10 m height. What is the angle of elevation of the top?” Height = 10 m (vertical side). Distance = 15 m (horizontal side). Formula used: tan(θ)= Distance/Height =10/15 This shows how both terms appear together in most height-and-distance problems. Everyday Relevance Architects measure the height of buildings but need the distance from roads to design sightlines. Pilots calculate the height of descent and distance to the runway simultaneously. Surveyors use angles to measure the height of hills from a certain distance.

Height and depth describe vertical displacement in opposite directions, but both are lengths. Since the underlying physical quantity is the same—length—scientists, engineers, and exam problems use the same units (meters, centimeters, feet) for both. A unified unit system keeps calculations consistent, comparable, and simple. 1. Core definitions and axis thinking Height: vertical measurement from a base reference upward to the top of an object. Example: a building with 60 m height. Depth : vertical measurement from a surface reference downward into a medium (ground, water). Example: a well of 40 m depth. In coordinate terms, treat the vertical axis as positive upward and negative downward. Both height and depth are magnitudes along that axis. Their sign or direction differs, not their dimension. 2. Why the same units in practice Dimensional consistency: Both are lengths, so the same unit preserves dimensional integrity across equations. Simpler calculations : Trigonometry frequently mixes vertical and horizontal measures. A single unit avoids conversion errors. Comparability : You can directly compare a mountain’s height with a trench’s depth when both are in meters. Standards : The SI system defines the meter as the global base unit for length, used across education, research, and industry. 3. Real-world examples Built environment: A basement 5 m below grade and a roof 15 m above are both measured in meters. Total vertical span = 20 m. Earth science : Everest at 8,848 m above sea level and the Mariana Trench ~11,000 m below sea level are routinely compared. Maritime and pools : A pier’s deck height and the water’s depth under it are both in meters or feet for safety calculations. 4. Implications for trigonometry and exam prep Right-triangle setups often treat depth as a negative height relative to a datum. Angles of depression align with downward verticals, but your numeric magnitudes stay in the same units, streamlining tan , sin , and cos applications and error checking. Key takeaways: Height ↑ and depth ↓ are both lengths along the vertical axis. Use the same units to maintain consistency, accuracy, and comparability. This convention simplifies trigonometric calculations and real-world measurement.

Height and depth describe vertical displacement in opposite directions, but both are lengths. Since the underlying physical quantity is the same—length—scientists, engineers, and exam problems use the same units (meters, centimeters, feet) for both. A unified unit system keeps calculations consistent, comparable, and simple. 1. Core definitions and axis thinking Height: vertical measurement from a base reference upward to the top of an object. Example: a building with 60 m height. Depth : vertical measurement from a surface reference downward into a medium (ground, water). Example: a well of 40 m depth. In coordinate terms, treat the vertical axis as positive upward and negative downward. Both height and depth are magnitudes along that axis. Their sign or direction differs, not their dimension. 2. Why the same units in practice Dimensional consistency: Both are lengths, so the same unit preserves dimensional integrity across equations. Simpler calculations : Trigonometry frequently mixes vertical and horizontal measures. A single unit avoids conversion errors. Comparability : You can directly compare a mountain’s height with a trench’s depth when both are in meters. Standards : The SI system defines the meter as the global base unit for length, used across education, research, and industry. 3. Real-world examples Built environment: A basement 5 m below grade and a roof 15 m above are both measured in meters. Total vertical span = 20 m. Earth science : Everest at 8,848 m above sea level and the Mariana Trench ~11,000 m below sea level are routinely compared. Maritime and pools : A pier’s deck height and the water’s depth under it are both in meters or feet for safety calculations. 4. Implications for trigonometry and exam prep Right-triangle setups often treat depth as a negative height relative to a datum. Angles of depression align with downward verticals, but your numeric magnitudes stay in the same units, streamlining tan , sin , and cos applications and error checking. Key takeaways: Height ↑ and depth ↓ are both lengths along the vertical axis. Use the same units to maintain consistency, accuracy, and comparability. This convention simplifies trigonometric calculations and real-world measurement.

We use meters for any length—heights, distances, depths—because the meter is the SI base unit for length and scales cleanly from millimeters to kilometers. We use radians for angles because many fundamental trigonometric identities and calculus limits are exact and elegant in radians, which are defined from circle geometry. 1. Why meters for length? Global standard: The SI system ensures consistent teaching, research, and industry usage. Scalability : Convert easily across mm–cm–m–km without changing the base concept. Interoperability : Diagrams, instruments, and datasets align on the same unit, reducing cognitive load and mistakes. 2. Why radians for angles? Natural definition : One radian is the angle subtending an arc equal to the radius. This ties angle directly to length on a circle. Mathematical simplicity : Identities and series like sin x ≈ x for small x are exact in radians, making derivations clean. Calculus compatibility : The limit lim x→0 sin x / x = 1 holds in radians, undergirding differential rules used across physics and engineering. 3. Degrees vs radians in learning Degrees are intuitive for common angles (30°, 45°, 60°) and quick mental geometry. Radians dominate in advanced topics where exactness and compact formulas matter. Exams may specify degrees for numeric problems but often expect radian fluency for proofs and calculus-linked questions. 4. Examples you’ll actually use Heights and distances: A tower of 20 m and a baseline of 35 m fit naturally with angles later converted to or computed in radians when needed. Rotational motion : Angular velocity and acceleration are inherently radian-based, simplifying units in kinematics. Waves and oscillations : Sine and cosine arguments are in radians, keeping phase and frequency relationships consistent. Key takeaways: Meters = universal, scalable unit for any length in trigonometry problems. Radians = natural angle unit for exact identities and calculus. Use degrees for intuition, radians for derivations and physics.

We use meters for any length—heights, distances, depths—because the meter is the SI base unit for length and scales cleanly from millimeters to kilometers. We use radians for angles because many fundamental trigonometric identities and calculus limits are exact and elegant in radians, which are defined from circle geometry. 1. Why meters for length? Global standard: The SI system ensures consistent teaching, research, and industry usage. Scalability : Convert easily across mm–cm–m–km without changing the base concept. Interoperability : Diagrams, instruments, and datasets align on the same unit, reducing cognitive load and mistakes. 2. Why radians for angles? Natural definition : One radian is the angle subtending an arc equal to the radius. This ties angle directly to length on a circle. Mathematical simplicity : Identities and series like sin x ≈ x for small x are exact in radians, making derivations clean. Calculus compatibility : The limit lim x→0 sin x / x = 1 holds in radians, undergirding differential rules used across physics and engineering. 3. Degrees vs radians in learning Degrees are intuitive for common angles (30°, 45°, 60°) and quick mental geometry. Radians dominate in advanced topics where exactness and compact formulas matter. Exams may specify degrees for numeric problems but often expect radian fluency for proofs and calculus-linked questions. 4. Examples you’ll actually use Heights and distances: A tower of 20 m and a baseline of 35 m fit naturally with angles later converted to or computed in radians when needed. Rotational motion : Angular velocity and acceleration are inherently radian-based, simplifying units in kinematics. Waves and oscillations : Sine and cosine arguments are in radians, keeping phase and frequency relationships consistent. Key takeaways: Meters = universal, scalable unit for any length in trigonometry problems. Radians = natural angle unit for exact identities and calculus. Use degrees for intuition, radians for derivations and physics.

Yes. You can estimate or compute height without measuring angles by using shadow ratios and similar triangles, direct instruments and sensors, or physics-based relations. The best method depends on accuracy needs and what information or tools you have. 1. Shadow ratios and similar triangles When the Sun’s rays are effectively parallel, two objects create similar right triangles. If a 1 m stick casts a 2 m shadow while a nearby tree casts a 10 m shadow, then: height_stick / shadow_stick = height_tree / shadow_tree1 / 2 = h / 10 → h = 5 m This approach avoids explicit angle measurement while leveraging proportionality. For better accuracy, measure at midday when shadows are sharp and the ground is level. 2. Direct measurement tools Laser distance meters: Some devices compute height using two distance readings and internal sensors, or by vertical scan if accessible. Altimeters and barometric sensors : Estimate altitude differences, useful in outdoor fieldwork. GNSS/GPS : Provides elevation data. Accuracy varies with device, satellite geometry, and environment. These methods are practical when you can reach the base or top or when line-of-sight tools are permitted. 3. Physics-based relations Free fall timing: If an object is dropped from rest and takes t seconds to reach the ground, h = ½ g t² . Use with caution and safety in controlled settings. Projectile or sound timing : In some setups, measure time-of-flight of a signal and infer height from known speeds. 4. Choosing the right approach No instruments: Use shadow ratios with a known reference stick. Field measurement : Use a laser meter or smartphone sensors if precision matters. Educational demo : Physics timing methods illustrate concepts but carry uncertainty and safety considerations. Key takeaways: You can compute height without angles using ratios, instruments, or physics. Accuracy depends on surface levelness, measurement quality, and method. Document assumptions, units, and uncertainties for dependable results.

Yes. You can estimate or compute height without measuring angles by using shadow ratios and similar triangles, direct instruments and sensors, or physics-based relations. The best method depends on accuracy needs and what information or tools you have. 1. Shadow ratios and similar triangles When the Sun’s rays are effectively parallel, two objects create similar right triangles. If a 1 m stick casts a 2 m shadow while a nearby tree casts a 10 m shadow, then: height_stick / shadow_stick = height_tree / shadow_tree1 / 2 = h / 10 → h = 5 m This approach avoids explicit angle measurement while leveraging proportionality. For better accuracy, measure at midday when shadows are sharp and the ground is level. 2. Direct measurement tools Laser distance meters: Some devices compute height using two distance readings and internal sensors, or by vertical scan if accessible. Altimeters and barometric sensors : Estimate altitude differences, useful in outdoor fieldwork. GNSS/GPS : Provides elevation data. Accuracy varies with device, satellite geometry, and environment. These methods are practical when you can reach the base or top or when line-of-sight tools are permitted. 3. Physics-based relations Free fall timing: If an object is dropped from rest and takes t seconds to reach the ground, h = ½ g t² . Use with caution and safety in controlled settings. Projectile or sound timing : In some setups, measure time-of-flight of a signal and infer height from known speeds. 4. Choosing the right approach No instruments: Use shadow ratios with a known reference stick. Field measurement : Use a laser meter or smartphone sensors if precision matters. Educational demo : Physics timing methods illustrate concepts but carry uncertainty and safety considerations. Key takeaways: You can compute height without angles using ratios, instruments, or physics. Accuracy depends on surface levelness, measurement quality, and method. Document assumptions, units, and uncertainties for dependable results.

You form a right triangle with the object’s height as the vertical side, your distance as the horizontal side, and your line of sight as the hypotenuse. Using tangent, sine, or cosine, you calculate the unknown side. 1. The basic idea When observing an object, your line of sight forms a triangle. Height = vertical side, distance = horizontal side, hypotenuse = line of sight. Trigonometric ratios connect these sides with the angle observed. 2. Core ratios sin θ = opposite / hypotenuse cos θ = adjacent / hypotenuse tan θ = opposite / adjacent tan θ is most common in height and distance problems. 3. Example Standing 40 m from a tower, angle of elevation = 45°. Then: tan 45° = h / 401 = h / 40 → h = 40 m The tower is 40 m tall. 4. Why it’s useful Practical for tall or far objects. Accurate with minimal equipment. Applicable in exams and real-world measurement. Important Note: Use trigonometric ratios with known angles and distances. tan is most common for height-distance relations. Method applies to buildings, mountains, trees, and more.

You form a right triangle with the object’s height as the vertical side, your distance as the horizontal side, and your line of sight as the hypotenuse. Using tangent, sine, or cosine, you calculate the unknown side. 1. The basic idea When observing an object, your line of sight forms a triangle. Height = vertical side, distance = horizontal side, hypotenuse = line of sight. Trigonometric ratios connect these sides with the angle observed. 2. Core ratios sin θ = opposite / hypotenuse cos θ = adjacent / hypotenuse tan θ = opposite / adjacent tan θ is most common in height and distance problems. 3. Example Standing 40 m from a tower, angle of elevation = 45°. Then: tan 45° = h / 401 = h / 40 → h = 40 m The tower is 40 m tall. 4. Why it’s useful Practical for tall or far objects. Accurate with minimal equipment. Applicable in exams and real-world measurement. Important Note: Use trigonometric ratios with known angles and distances. tan is most common for height-distance relations. Method applies to buildings, mountains, trees, and more.

Trigonometry is used daily in engineering, navigation, surveying, and construction to estimate heights and distances indirectly by measuring angles and baselines. 1. Everyday applications Architecture: Building height estimation and design. Navigation : Aircraft altitude calculations. Forestry : Measuring tree heights. Surveying : Mapping terrain without direct distance travel. 2. Example Forester stands 25 m from a tree, angle of elevation = 30°: tan 30° = h / 25 0.577 = h / 25 → h = 14.4 m The tree is 14.4 m tall. 3. Broader use Satellites: Align antennas by elevation angles. Civil engineering : Bridge span design. Astronomy : Calculating celestial distances. 4. Why explanations must help Connecting formulas to relatable tasks—like tree measurement—makes trigonometry memorable and practical, especially for exam prep and STEM careers. Important Note: Trigonometry is applied in many professions and sciences. Angles of elevation and depression are central. Real-world problems make the learning engaging and relevant.

Trigonometry is used daily in engineering, navigation, surveying, and construction to estimate heights and distances indirectly by measuring angles and baselines. 1. Everyday applications Architecture: Building height estimation and design. Navigation : Aircraft altitude calculations. Forestry : Measuring tree heights. Surveying : Mapping terrain without direct distance travel. 2. Example Forester stands 25 m from a tree, angle of elevation = 30°: tan 30° = h / 25 0.577 = h / 25 → h = 14.4 m The tree is 14.4 m tall. 3. Broader use Satellites: Align antennas by elevation angles. Civil engineering : Bridge span design. Astronomy : Calculating celestial distances. 4. Why explanations must help Connecting formulas to relatable tasks—like tree measurement—makes trigonometry memorable and practical, especially for exam prep and STEM careers. Important Note: Trigonometry is applied in many professions and sciences. Angles of elevation and depression are central. Real-world problems make the learning engaging and relevant.

Ancient engineers used geometry, shadows, and simple instruments like gnomons and astrolabes to apply similar triangles and ratios to height measurement. 1. Greek contributions Thales of Miletus measured pyramid height by comparing the ratio of its shadow to a stick’s shadow—an early similar triangle application. 2. Indian and Islamic advances Indian mathematicians like Aryabhata applied trigonometry to astronomy. Islamic scholars refined sine and cosine for surveying and navigation. 3. Instruments used Gnomon: Vertical stick to cast measurable shadows. Astrolabe : Angle-measuring device for stars and elevation. Quadrant : Tool for measuring angles of elevation and depression. 4. Example Pyramid’s height estimated by shadow ratio: height_stick / shadow_stick = height_pyramid / shadow_pyramid This method mirrors modern similar triangle problems taught in classrooms. 5) Modern relevance These ancient methods highlight how practical measurement needs spurred mathematical development. They remain instructive and demonstrate trigonometry’s timeless relevance. Important: Ancient engineers used geometry and instruments for height measurement. Shadow ratios and similar triangles were primary methods. Understanding history enriches modern learning of trigonometry.

Ancient engineers used geometry, shadows, and simple instruments like gnomons and astrolabes to apply similar triangles and ratios to height measurement. 1. Greek contributions Thales of Miletus measured pyramid height by comparing the ratio of its shadow to a stick’s shadow—an early similar triangle application. 2. Indian and Islamic advances Indian mathematicians like Aryabhata applied trigonometry to astronomy. Islamic scholars refined sine and cosine for surveying and navigation. 3. Instruments used Gnomon: Vertical stick to cast measurable shadows. Astrolabe : Angle-measuring device for stars and elevation. Quadrant : Tool for measuring angles of elevation and depression. 4. Example Pyramid’s height estimated by shadow ratio: height_stick / shadow_stick = height_pyramid / shadow_pyramid This method mirrors modern similar triangle problems taught in classrooms. 5) Modern relevance These ancient methods highlight how practical measurement needs spurred mathematical development. They remain instructive and demonstrate trigonometry’s timeless relevance. Important: Ancient engineers used geometry and instruments for height measurement. Shadow ratios and similar triangles were primary methods. Understanding history enriches modern learning of trigonometry.

CBSE Class 12 Maths Notes on Heights and Distances – Concepts, Formulas & Applications

Angle of Elevation

If 'O' be the observer's eye and OX be the horizontal line through O. If the object P is at a higher level than eye, then angle POX = θ is called the angle of elevation.

Angle of Elevation

Angle of Depression

If the object P is at a lower level than O, then angle POX is called the angle of depression.

Angle of Depression

Illustration 1: A ladder leaning against a vertical wall is inclined at an angle α to the horizontal. On moving its foot 2 m away from the wall, the ladder is now inclined at angle β. Then the vertical distance moved by the ladder is

(A) 2 sin β/(sin β – sin α)

(B) 2 cos β/(sin β – sin α)

(D) 2 cos β/(sin β + sin α)

Solution: (B)

From triangle ABC:

l = 2 sec α ...(1)

x + d = 2 tan α ...(2)

From triangle CDE:

x = l sin β ...(3)

From (1) and (3): x = 2 sec α sin β

Putting in (2): 2 sec α sin β + d = 2 tan α

d = 2 tan α – 2 sec α sin β

d = (2 sin α/cos α) – (2 sin β/cos α)

d = (2/cos α)(sin α – sin β)

Therefore: d = (2/cos β)(sin β – sin α)

Illustration 2: The altitude of a rock is observed to be 47°. After walking 1000 m towards it, up a slope inclined at 32° to the horizon, the altitude is 77°. Then the vertical height of the rock above the first point of observation is (given that sin 47° = 0.73)

(A) 730√2 m (B) 730 m (C) 730/√2 m (D) none of these

Solution: (A)

In triangle ABD:

l/1000 = sin 30°/sin 45°

l = 2000 × (1/2) = 1000√2

AC = l sin 47°

AC = 1000√2 × 0.73 = 730√2 m

vertical height of the rock above the first point of observation

Exercise Problems:

(i) A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60°. When he retires 40 feet from the bank, he finds the angle to be 30°. Then the height of the tree is

(A) 20 ft

(B) 20√3 ft

(D) 20√2 ft

(ii) The angular depressions of the top and the foot of a chimney as seen from the top of a second chimney which is 150 meters high and standing on the same level as the first are α and β respectively. Then the distance between their tops, when tan α = 4/3, tan β = 5/2 is

(A) 100 m (B) 120 m (C) 110 m (D) none of these

Exercise 1 Answers:

(i) B

(ii) A

Formulas and Concepts at a Glance

Key Concepts:

If 'O' be the observer's eye and OX be the horizontal line through O. If the object P is at a higher level than eye, then angle POX = θ is called the angle of elevation
If the object P is at a lower level than O, then angle POX is called the angle of depression.

Solved Examples

1. Length of the shadow of a person is x when the angle of elevation of the sun is 45°. If the length of the shadow increases by (√3 – 1)x, then the angle of elevation of sun should become

(A) 15° (B) 18° (C) 25° (D) 30°

Solution: (D) Let h be the height of the person

h/x = tan 45° = 1 ⟹ h = x

Now, tan θ = h/[x + (√3 – 1)x] = x/(√3x) = 1/√3

Therefore θ = 30°

Let h be the height of the person

2. At a certain point the angle of elevation of a tower is found to be such that its cotangent is 3/5; on walking 32 m directly towards the tower its angle of elevation is an angle whose cotangent is 2/3. Then the height of tower is

(A) 120 m (B) 80 m (C) 160 m (D) 210 m

Solution: (C)

AB = tower

cot ∠ADB = 3/5 ⟹ BD/AB = 3/5 ⟹ (32 + BC)/AB = 3/5

AB = (5/3)(32 + BC) ...(1)

cot ∠ACB = 2/3 ⟹ BC/AB = 2/3 ⟹ (2/3)BC = AB ...(2)

From (1) and (2): (2/3)BC = (5/3)(32 + BC) ⟹ BC = 64 m

AB = (5/2) × 64 = 160 m

the height of tower is

3. A light house, facing north, sends out a fan shaped beam of light extending from NE (north east) to NW (north west). An observer on a steamer, sailing due west, first sees the light when he is 5 km away from the light house and continues to see it for 30√2 minutes. Then speed of the steamer is

(A) 5 km/h

(B) 15 km/h

(D) 10 km/h

Solution: (D)

Light house is at A and it throws the light from direction AB to AC.

AB = AC = 5 km, ∠BAC = 90°

BC = 5√2 km

Speed of streamer = (5√2)/(30√2/60) = 10 km/h

Then speed of the steamer is

4. The angle of elevation of the top of a T.V. tower from three points A, B, C in a straight line in the horizontal plane through the foot of the tower are α, 2α, 3α respectively. If AB = a, the height of the tower is

(A) a tan α (B) a sin α (C) a sin 2α (D) a sin 3α

Solution: (C)

AB = a, BP = a from isosceles triangle ABP

h = BP sin 2α = a sin 2α

ngle of elevation of the top of a T.V. tower from three points

5. A man on the top of a cliff 100 meter high, observes the angles of depression of two points on the opposite sides of the cliff as 30° and 60° respectively. Find the distance between the two points.

(A) 400/√3 meters (B) 100√3 meter (C) 400 meter (D) none of these

Solution: (A)

Let PQ be the cliff and A and B be the points under observation.

PQ = 100 meters

AP = 100 cot 30° = 100√3

BP = 100 cot 60° = 100/√3

Hence AB = AP + BP = 100√3 + 100/√3 = 400/√3 m

6. A flag–staff of 6 m high placed on the top of a tower throws a shadow of 2√3 m along the ground. Then the angle that the sun makes with the ground is:

60°
30°
45°
none of these

Solution (Answer: A)

Let the tower’s height be h. Let the tower’s shadow be x. The sun makes an angle θ with the ground.

tan θ = h / x and tan θ = (h + 6) / (x + 2√3)

⇒ h(x + 2√3) = x(h + 6) ⇒ 2√3 h = 6x ⇒ x = h / √3

tan θ = h / x = √3 ⇒ θ = 60°.

Angle with the ground = 60°

tower’s shadow be x

7. ABC (right-angled at A), the hypotenuse is 2√2 times the length of the perpendicular from A to the hypotenuse. Then the angles B and C are:

67.5° and 22.5°
68.5° and 21.5°
70° and 20°
none of these

Answer: A

Let L be the foot of the perpendicular from A to the hypotenuse BC. Then BC = 2√2 · AL, and

BL/AL + LC/AL = BC/AL ⇒ cot B + cot C = 2√2.

Since B + C = 90°, we have cot C = tan B. Hence,

cot B + tan B = 2√2 ⇒ 2 csc(2B) = 2√2 ⇒ csc(2B) = √2.

2B = 45° or 135° ⇒ B = 22.5° or 67.5°; then C = 67.5° or 22.5°.

Angles: B = 67.5°, C = 22.5° (or vice-versa)

Let L be the foot of the perpendicular from A to the hypotenuse BC

8. A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60°. When he returns 100 m from the bank, he finds the angle to be 30°. Then the height of the tree is:

80√3 m
60√3 m
80 m
50√3 m

Solution (Answer: D)

Let AB = h (height of tree), CB = x (breadth of river). ∠BCA = 60°. Let D be second position of man so that ∠BDA = 30°.

tan 60° = h/x ⇒ h = √3x ...(1)

tan 30° = h / (100 + x) = 1/√3 ⇒ h = (100 + x)/√3 ...(2)

From (1) and (2): √3(√3x) = 100 + x ⇒ 3x − x = 100 ⇒ x = 50. So, h = 50√3.

Height of the tree = 50√3 m

angle to be 30°

9. AB is a vertical pole. A is on the ground. C is the midpoint of AB. P is a point on the ground. The portion CB subtends angle β at P. If AP = n·AB, then tan β is:

n / (2n² + 1)
n / (2n² − 1)
n / (n² + 1)
none of these

Solution (Answer: A)

Given AP = n·AB.

∠APB = α, ∠APC = α − β

From △APB: tan α = AB / AP = 1/n

From △APC: tan(α − β) = AC / AP = (1/2·AB) / (n·AB) = 1/(2n)

So tan β = tan α − tan(α − β) / (1 + tan α·tan(α − β))

= (1/n − 1/(2n)) / (1 + (1/n)(1/(2n)))
= (1/(2n)) / ((2n² + 1)/(2n²))
= n / (2n² + 1)

tan β = n / (2n² + 1)

AP = n·AB, then tan β is

Frequently Asked Questions

The difference between length, height, and distance is fundamental in mathematics and real-life problem solving. These three terms are often used interchangeably in casual conversation, but in geometry, physics, and trigonometry, each has a specific meaning. Let’s break them down clearly.

Length
- Length is the measurement of something from end to end.
- It typically refers to the longest side of an object or the measurement along a straight line.
- Example: The length of a pencil is 15 cm.
In mathematics, when we say “length of a line segment,” we mean the numerical value that tells us how far apart its endpoints are. Length can be horizontal, vertical, or diagonal depending on the orientation.
Height
- Height specifically refers to the vertical measurement from the bottom (base) to the top.
- It is always measured perpendicular to the ground or a reference base.
- Example: The height of a building is 50 meters.
Height becomes important in trigonometry when we calculate the angle of elevation or depression. Whenever you see a problem involving towers, trees, or poles, the “height” is the vertical side of the right triangle.
Distance
- Distance refers to the separation between two points in space.
- It may be along a straight line (shortest path) or along a curved path.
- Example: The distance between two towns is 25 km.
In trigonometry, distance usually refers to the horizontal distance between an observer and the object being viewed. It is the base of the right triangle formed with height and the line of sight.

Practical Example

Imagine you are standing 30 meters away from a flagpole.

The height is the vertical measurement of the flagpole, say 20 meters.
The distance between you and the base of the flagpole is 30 meters.
The length could refer to the size of the flag itself, say 5 meters from one end to another.

Thus, in one scenario, all three terms can appear but have very different meanings.

Why does this matter for students?

Students preparing for exams often confuse these terms. Misunderstanding them can lead to incorrect identification of triangle sides in trigonometric problems. To avoid errors:

Always identify height as vertical.
Always identify distance as the horizontal ground separation.
Use length in a general sense for measuring objects or lines.

The terms distance and height are central in trigonometry and geometry, yet they often cause confusion. Let’s make a clear distinction.

Height
- Defined as the vertical measurement from base to top.
- Always measured perpendicular to the ground or reference line.
- Example: The height of a streetlight is 10 meters.
In trigonometry, height is the opposite side when calculating with sine, cosine, or tangent ratios.
Distance
- Refers to the horizontal ground measurement between two points.
- Example: The distance from a person to the base of the streetlight is 15 meters.
In trigonometry, distance usually represents the adjacent side of the right triangle.

Trigonometric Context

Suppose a student is asked: “A person is standing 15 m away from a streetlight of 10 m height. What is the angle of elevation of the top?”

Height = 10 m (vertical side).
Distance = 15 m (horizontal side).
Formula used: tan(θ)= Distance/Height =10/15

This shows how both terms appear together in most height-and-distance problems.

Everyday Relevance

Architects measure the height of buildings but need the distance from roads to design sightlines.
Pilots calculate the height of descent and distance to the runway simultaneously.
Surveyors use angles to measure the height of hills from a certain distance.

Height and depth describe vertical displacement in opposite directions, but both are lengths. Since the underlying physical quantity is the same—length—scientists, engineers, and exam problems use the same units (meters, centimeters, feet) for both. A unified unit system keeps calculations consistent, comparable, and simple.

1. Core definitions and axis thinking

Height: vertical measurement from a base reference upward to the top of an object. Example: a building with 60 m height.
Depth: vertical measurement from a surface reference downward into a medium (ground, water). Example: a well of 40 m depth.

In coordinate terms, treat the vertical axis as positive upward and negative downward. Both height and depth are magnitudes along that axis. Their sign or direction differs, not their dimension.

2. Why the same units in practice

Dimensional consistency: Both are lengths, so the same unit preserves dimensional integrity across equations.
Simpler calculations: Trigonometry frequently mixes vertical and horizontal measures. A single unit avoids conversion errors.
Comparability: You can directly compare a mountain’s height with a trench’s depth when both are in meters.
Standards: The SI system defines the meter as the global base unit for length, used across education, research, and industry.

3. Real-world examples

Built environment: A basement 5 m below grade and a roof 15 m above are both measured in meters. Total vertical span = 20 m.
Earth science: Everest at 8,848 m above sea level and the Mariana Trench ~11,000 m below sea level are routinely compared.
Maritime and pools: A pier’s deck height and the water’s depth under it are both in meters or feet for safety calculations.

4. Implications for trigonometry and exam prep

Right-triangle setups often treat depth as a negative height relative to a datum. Angles of depression align with downward verticals, but your numeric magnitudes stay in the same units, streamlining tan, sin, and cos applications and error checking.

Key takeaways:

Height ↑ and depth ↓ are both lengths along the vertical axis.
Use the same units to maintain consistency, accuracy, and comparability.
This convention simplifies trigonometric calculations and real-world measurement.

We use meters for any length—heights, distances, depths—because the meter is the SI base unit for length and scales cleanly from millimeters to kilometers. We use radians for angles because many fundamental trigonometric identities and calculus limits are exact and elegant in radians, which are defined from circle geometry.

1. Why meters for length?

Global standard: The SI system ensures consistent teaching, research, and industry usage.
Scalability: Convert easily across mm–cm–m–km without changing the base concept.
Interoperability: Diagrams, instruments, and datasets align on the same unit, reducing cognitive load and mistakes.

2. Why radians for angles?

Natural definition: One radian is the angle subtending an arc equal to the radius. This ties angle directly to length on a circle.
Mathematical simplicity: Identities and series like sin x ≈ x for small x are exact in radians, making derivations clean.
Calculus compatibility: The limit lim_x→0 sin x / x = 1 holds in radians, undergirding differential rules used across physics and engineering.

3. Degrees vs radians in learning

Degrees are intuitive for common angles (30°, 45°, 60°) and quick mental geometry. Radians dominate in advanced topics where exactness and compact formulas matter. Exams may specify degrees for numeric problems but often expect radian fluency for proofs and calculus-linked questions.

4. Examples you’ll actually use

Heights and distances: A tower of 20 m and a baseline of 35 m fit naturally with angles later converted to or computed in radians when needed.
Rotational motion: Angular velocity and acceleration are inherently radian-based, simplifying units in kinematics.
Waves and oscillations: Sine and cosine arguments are in radians, keeping phase and frequency relationships consistent.

Key takeaways:

Meters = universal, scalable unit for any length in trigonometry problems.
Radians = natural angle unit for exact identities and calculus.
Use degrees for intuition, radians for derivations and physics.

Yes. You can estimate or compute height without measuring angles by using shadow ratios and similar triangles, direct instruments and sensors, or physics-based relations. The best method depends on accuracy needs and what information or tools you have.

1. Shadow ratios and similar triangles

When the Sun’s rays are effectively parallel, two objects create similar right triangles. If a 1 m stick casts a 2 m shadow while a nearby tree casts a 10 m shadow, then:

height_stick / shadow_stick = height_tree / shadow_tree
1 / 2 = h / 10 → h = 5 m

This approach avoids explicit angle measurement while leveraging proportionality. For better accuracy, measure at midday when shadows are sharp and the ground is level.

2. Direct measurement tools

Laser distance meters: Some devices compute height using two distance readings and internal sensors, or by vertical scan if accessible.
Altimeters and barometric sensors: Estimate altitude differences, useful in outdoor fieldwork.
GNSS/GPS: Provides elevation data. Accuracy varies with device, satellite geometry, and environment.

These methods are practical when you can reach the base or top or when line-of-sight tools are permitted.

3. Physics-based relations

Free fall timing: If an object is dropped from rest and takes t seconds to reach the ground, h = ½ g t². Use with caution and safety in controlled settings.
Projectile or sound timing: In some setups, measure time-of-flight of a signal and infer height from known speeds.

4. Choosing the right approach

No instruments: Use shadow ratios with a known reference stick.
Field measurement: Use a laser meter or smartphone sensors if precision matters.
Educational demo: Physics timing methods illustrate concepts but carry uncertainty and safety considerations.

Key takeaways:

You can compute height without angles using ratios, instruments, or physics.
Accuracy depends on surface levelness, measurement quality, and method.
Document assumptions, units, and uncertainties for dependable results.

You form a right triangle with the object’s height as the vertical side, your distance as the horizontal side, and your line of sight as the hypotenuse. Using tangent, sine, or cosine, you calculate the unknown side.

1. The basic idea

When observing an object, your line of sight forms a triangle. Height = vertical side, distance = horizontal side, hypotenuse = line of sight. Trigonometric ratios connect these sides with the angle observed.

2. Core ratios

sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent

tan θ is most common in height and distance problems.

3. Example

Standing 40 m from a tower, angle of elevation = 45°. Then:

tan 45° = h / 40
1 = h / 40 → h = 40 m

The tower is 40 m tall.

4. Why it’s useful

Practical for tall or far objects.
Accurate with minimal equipment.
Applicable in exams and real-world measurement.

Important Note:

Use trigonometric ratios with known angles and distances.
tan is most common for height-distance relations.
Method applies to buildings, mountains, trees, and more.

Trigonometry is used daily in engineering, navigation, surveying, and construction to estimate heights and distances indirectly by measuring angles and baselines.

1. Everyday applications

Architecture: Building height estimation and design.
Navigation: Aircraft altitude calculations.
Forestry: Measuring tree heights.
Surveying: Mapping terrain without direct distance travel.

2. Example

Forester stands 25 m from a tree, angle of elevation = 30°:

tan 30° = h / 25 0.577 = h / 25 → h = 14.4 m

The tree is 14.4 m tall.

3. Broader use

Satellites: Align antennas by elevation angles.
Civil engineering: Bridge span design.
Astronomy: Calculating celestial distances.

4. Why explanations must help

Connecting formulas to relatable tasks—like tree measurement—makes trigonometry memorable and practical, especially for exam prep and STEM careers.

Important Note:

Trigonometry is applied in many professions and sciences.
Angles of elevation and depression are central.
Real-world problems make the learning engaging and relevant.

Ancient engineers used geometry, shadows, and simple instruments like gnomons and astrolabes to apply similar triangles and ratios to height measurement.

1. Greek contributions

Thales of Miletus measured pyramid height by comparing the ratio of its shadow to a stick’s shadow—an early similar triangle application.

2. Indian and Islamic advances

Indian mathematicians like Aryabhata applied trigonometry to astronomy. Islamic scholars refined sine and cosine for surveying and navigation.

3. Instruments used

Gnomon: Vertical stick to cast measurable shadows.
Astrolabe: Angle-measuring device for stars and elevation.
Quadrant: Tool for measuring angles of elevation and depression.

4. Example

Pyramid’s height estimated by shadow ratio:

height_stick / shadow_stick = height_pyramid / shadow_pyramid

This method mirrors modern similar triangle problems taught in classrooms.

5) Modern relevance

These ancient methods highlight how practical measurement needs spurred mathematical development. They remain instructive and demonstrate trigonometry’s timeless relevance.

Important:

Ancient engineers used geometry and instruments for height measurement.
Shadow ratios and similar triangles were primary methods.
Understanding history enriches modern learning of trigonometry.

Measure the distance from the building, find the angle of elevation, and apply the tangent formula. Add observer’s eye level for full accuracy.

1) Principle

Height forms the opposite side, distance is the adjacent side, and the line of sight is the hypotenuse of a right triangle. Tangent connects height and distance.

2) Formula

tan θ = height / distance

3) Example

At 30 m away, angle of elevation = 60°, eye level = 1.6 m:

tan 60° = h / 30
√3 = h / 30
h = 51.96 m
Total height = 51.96 + 1.6 = 53.56 m

4) Applications

Construction engineers use it for height verification.
Fire departments calculate ladder requirements.
Students solve such problems in competitive exams.

Important Note:

Use tangent formula with measured distance and angle.
Always add observer eye-level for accuracy.
Practical in surveying, construction, and learning contexts.

Use measured angles from a known baseline and apply the law of cosines or sines to calculate the distance between peaks.

1. Principle

Surveyors set up at point A, measure angles to mountain peaks B and C, and use triangle ABC for calculations.

2. Formula (Law of Cosines)

BC² = AB² + AC² - 2·AB·AC·cos ∠A

3. Example

AB = 2 km, AC = 3 km, ∠A = 40°:

BC² = 2² + 3² - 2·2·3·cos40°
BC² ≈ 3.808
BC ≈ 1.95 km

4. Applications

Geologists measure spacing between mountain ranges.
Military maps valleys and peaks for navigation.
Pilots assess valley width for safe passage.

Note:

Measure angles from a baseline to apply cosine or sine laws.
Useful in geography, geology, and navigation.
Avoids physically traversing hazardous terrain.

Measure your distance from the tree, record the angle of elevation to its top, and apply the tangent formula. Add your eye-level height for the total tree height.

1. Principle

The tree height is the opposite side of a right triangle, your distance is the adjacent side. Tangent ratio links them.

2. Formula

tan θ = height / distance

3. Example

Distance = 20 m, angle = 30°, eye level = 1.5 m:

tan 30° = h / 20
0.577 = h / 20
h = 11.54 m
Total height = 11.54 + 1.5 = 13.04 m

4. Applications

Foresters estimate tree growth for logging or conservation.
Environmental researchers measure canopy heights.
Students practice this as standard trigonometry problems.

Note:

Use tangent with measured distance and angle.
Add eye-level correction for full height.
Widely applied in forestry, environment, and education.

S.no	Formulas List
1.	Trigonometric equations
2.	Solution of the triangle
3.	Height and distance
4.	Inverse trigonometric function
5.	Exponential and logarithm series
6.	Logarithm & properties
7.	Quadratic equations
8.	Indefinite integral
9.	Definite integrals
10.	Differential equation

Height and distance

Angle of Elevation

Angle of Depression

Exercise Problems:

Formulas and Concepts at a Glance

Frequently Asked Questions

Related Links

Become one of thousands of satisfied students!