Height and Distance: Definition, Formula and Sample Questions

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One of the real-life applications of trigonometry is Height and Distances and basically uses trigonometric ratios to calculate heights and lengths. Some important definitions that we would make use of while learning this concept are Line of sight, Angle of elevation, Angle of depression, etc. Let’s dive right in!

Read More: Trigonometry Table

Table of Content

Important Definitions
How to calculate Heights and Distances?
Things to Remember
Sample Questions

Read More: Eccentricity

Important Definitions

Some Important Definitions to understand the concepts of Height and Distance are as follows:

Line of Sight: The line which is drawn from the eyes of the observer to the point being viewed on the object is known as the line of sight.

Angle of Elevation: The angle of elevation of the point on the object (above horizontal level) viewed by the observer is the angle which is formed by the line of sight with the horizontal level.

Angle of Depression: The angle of depression of the point on the object (below horizontal level) viewed by the observer is the angle that is formed by the line of sight with the horizontal level.

Also Read:

Sin Squared x Formula with Solved Examples	Factorial Formula
Centroid of a Triangle	Algebra formulas

How to calculate Heights and Distances?

Trigonometric ratios are used to measure the heights and distances of objects. We use an example to illustrate. In the above picture, the line joining the observer’s eye and parallel to the ground is at the horizontal level and hence is called the Horizontal.

The line joining the observer’s eyes to the point of observation is called the Line of Sight as labelled in the diagram.

The Angle of Elevation is the angle between the horizontal and the Line of Sight.

Read More:

Types Of Triangles	Properties of Triangle	Formula Of Perimeter Shapes

Similarly, in a situation when the observer is higher than the point of observation, the angle is called the Angle of Depression.

Having studied Trigonometry, we know that the trigonometric ratios of angles in these triangles would be the same as the ratio of sides, and hence are a straightforward application of the ratios we learnt. Let’s quickly revise the values of these trigonometric ratios.

Because of the straightforward nature of the application, the types of problems can be reduced to a few common cases. These are as follows:

Case 1: In this case, two of the following 3 quantities would be given, and we use them to find out the third unknown quantity.

Height of the object (tower/building etc.)
Distance of an observer from the foot of the tower, hill, or building.
Angle of elevation or Angle of depression.

Angle of elevation or Angle of depression.

sin (x) = Opposite/Hypotenuse

cos (x) = Adjacent/Hypotenuse

tan (x) = Opposite/Hypotenuse

Read More: Differential Equation

Case 2: A whole lot of separate problems fall in this category. One of these is the movement of an observer. As the observer moves toward the object under observation, the angle of elevation increases. And similarly, it decreases when the observer moves away. The distance (d) moved by the observer can be calculated by the following formula.

d = h (cot x – cot y)

Case 3: This case sees the implementation of similar triangles. The triangles have a common angle and parallel sides.

As AB || ED, we can use one of the Similar triangles (Thales or BPT) theorems to calculate the unknown quantity.

AB/ED = BC/DC

Also Read:

Limit Formula	Surface Area of a Cone Formula
Midpoint Formula	Geometry Formula

Points to Remember

Following are some important points:

Assume the observer to be a point if his height is not given.
Angles of elevation and depression are always acute angles.
If the angle of elevation of sun decreases, then the length of shadow of an object increases and vice-versa.
If in problems, the angle of elevation of an object is given, then we conclude that the object is at higher altitude than observer. The angle of depression implies that observer is at higher altitude than object.

Read More:

Calculus Formula	Definite Integral Formula	Determinant Formula

Sample Questions

Ques: A ladder 15 m long makes an angle of 60^o with the wall. Find the height of the point where the ladder touches the wall.

Ans: cos 60^o = x/15

=>> ½ = x/15

=>> x = 15/2 m

=>> x = 7.5 m

The height of the point where the ladder touches the wall

Ques: An observer 1.5 m tall is 20.5 m away from a tower 22 m high. Determine the angle of elevation of the top of the tower from the eye of the observer.

Ans: PQ = MB = 1.5 m

Read More:

Transversal	Isosceles triangle	Scalene Triangle

AM = AB – MB = 22 – 1.5 = 20.5 m

Now in triangle APM,

Tan θ = AM/PM

=>> 20.5/20.5

=>> 1

=>> 45^o

The angle of elevation of the top of the tower from the eye of the observer.

Ques: AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of pole. If AD = 2.54 m, find the length of the ladder.

Ans: DB = (6 – 2.54) m = 3.46 m

Read More: Sin2x Formula

In triangle BDC, sin 60^o = BD/CD

=>> √3 / 2 = 3.46 / CD

=>> CD = 3.46*2 / 1.73

=>> DC = 4 m

Ques: If a tower 30 m high, casts a shadow 10√3 m long on the ground, then what is the angle of elevation of the sun?

Ans: Tan θ = AC/AB

=>> Tan θ = 30 / 10√3

=>> Tan θ = √3

=>> Tan θ = Tan 60^o

=>> θ = 60^o

Ques: From the top of a 10 m high building, the angle of elevation of a tower is 60° and the angle of depression of its foot at 45°. Determine the height of the tower.

Ans: Given: DE = 10m, ∠CDB = 45^o, ∠CDA = 60^o.

To find: AB

Solution: Tan ∠CDB = CB/CD

Tan 45^o = DE/CD ($\because$CB=DE)

=>> 1 = 10/CD

=>> CD = 10 m

Now, Tan ∠CDA = CA/CD

Tan 60^o = CA/10

=>> √3 = CA/10

=>> CA = 10√3

Now, AB = AC + BC

AB = 10 + 10√3 ≈ 27.32 m

It means that the height of the tower is 27.32 m.

Also Read:

Cosine Formula	Centroid Formula
Binomial Expansion Formula	Angle Formula

CBSE X Related Questions

1.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.
View Solution
2.
A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.
View Solution
3.
If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals
- $x^2 + 5x - 4$
- $(x + 3) (-x + 8)$
- $a(x^2 + 5x - 24)$
- $x^2 - 24$
View Solution
4.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.
View Solution
5.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is
- $50^\circ$
- $60^\circ$
- $45^\circ$
- $30^\circ$
View Solution
6.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.
View Solution

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Height and Distance: Definition, Formula and Sample Questions

Important Definitions

How to calculate Heights and Distances?

Points to Remember

Sample Questions

CBSE X Related Questions

1.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

2.
A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

3.
If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

4.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

5.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

6.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

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Height and Distance: Definition, Formula and Sample Questions

Important Definitions

How to calculate Heights and Distances?

Points to Remember

Sample Questions

CBSE X Related Questions

1.PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

2.A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

3.If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

4.A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

5.An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

6.In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

2.
A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

3.
If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

4.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

5.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

6.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.