Measurement of Angles Examples

An angle can be defined as the rotation from the initial point to an endpoint of a ray. Angle measurement is the amount of rotation from the initial to an endpoint of a ray. The angle is said to be a positive angle if the rotation is clockwise and negative if the rotation is anticlockwise. Angles can be measured by various units.

In this article, the degree, radian and grade measure are discussed, as they are the most commonly used units.

Types of Angles

How to Measure an Angle?

The different systems for angle measurement are as follows:

Measurement of Angle – Degree Measure

A degree is defined as a complete rotation in either a clockwise or anticlockwise direction, where the beginning and the ending point are the same. The rotation is divided into 360 units.

It is said to be 1⁰ if the rotation from the initial and ending side is 1 / 360^thof the rotation. The degree is divided into hours, minutes and seconds. One degree is 60 minutes, and one minute is 60 seconds.

Examples on Angle Measurement

Measurement of Angle – Radian Measure

Consider a circle of radius one unit. Let the arc of the circle be one unit. The measure of the angle is 1 radian if the arc subtends at the centre, provided the radius and arc lengths are equal.

The arc length of a circle with radius unity is equal to the angle in the radian.

Measurement of Angle – Grade Measure

A grade can be defined as a right angle divided into a hundred equal parts. Further, each grade is divided into a hundred minutes and each minute into a hundred seconds.

Measurement of Angles Formula

The following formulae can be used in the measurement of angles:

Degree Measure

\(\begin{array}{l}1^0=[\frac{1}{360}]^{th}\ \text{of a complete rotation}\end{array} \)

Radian Measure

θ = l/r, where l is the arc length, and r is the radius of the circle

Relation between Degree, Radian and Grade Measure

1) Degree and Radian Measure

It is known that a complete rotation in degrees is 360⁰, and 1 complete rotation = 2π radians in radian measure.

2π (radians) = 360° (degrees) or π (radians) = 180° (degrees)

Assuming, π = 3.14159

\(\begin{array}{l}1 \ radian= \frac{180^0}{\pi} \\ 1 \ degree = \frac{\pi}{180}\end{array} \)

Relation between degree and radian measure

Table of degree to radian measure of angles

2] Degree, Radian and Grade Measures

Let D be the number of degrees, R be the number of radians, and G be the number of grades in an angle θ, then

\(\begin{array}{l}\frac{D}{90}=\frac{G}{100}=\frac{2R}{\pi }\end{array} \)

This is the required relationship between the three systems of angle measurement.

Therefore, one radian

\(\begin{array}{l}\cos A.\cos 2A.\cos {{2}^{2}}A.\cos {{2}^{3}}A…….\cos {{2}^{n-1}}A=\frac{\sin {{2}^{n}}A}{{{2}^{n}}\sin A},\ \text{if }\ A=n\pi\ \text{radians} ={{180}^{o}}\\ i.e., 1\ \text{radian} =57{}^\circ 1{7}’44.{{8}’}’\approx {{57}^{o}}1{7}’4{{5}’}’.\end{array} \)

Example 1: Find the missing angles in the diagram and justify your answer.

Example on Missing Angles

Solution:

a = 105° because a = e (Corresponding angles)

b = 75° because b = 180° – a (Supplementary angles)

c = 105° because a = c (Opposite angles)

d = 75° because d = b (Opposite angles)

e = 105° because e = 105° (Opposite angles)

f = 75° because f = 180° – e (Supplementary angles)

h = 75° because h = 180° – 105° (Supplementary angles)

Example 2: Find the following missing angles.

Example for measuring angles

Solution:

Find angle a.

a = 35° because 80° + 65° + a = 180° (Supplementary angles)

a = 180° – 145°

or a = 35°

Find angle b.

b = 50° because 35° + 95° + b = 180° (Sum of angles in a triangle)

b = 180° – 130°

or b = 60°

Find for c.

c = 85° because c + 95° = 180° (Supplementary angles)

or c = 180° – 95° = 85°

Find for d.

d = 30° because 65° + 85° + c = 180° (Sum of angles in a triangle)

d = 180° – 150° = 30°

Example 3: Find the unknown angles from the figure below.

Examples Solution on Missing Angles

Solution:

The triangle given is an isosceles triangle. The two base angles will be equal.

Hence, a = 35°

The two angles of the triangle are known. Adding the third angle makes 180°.

35° + 35° + b = 180°

b = 180° – 70°

b = 110°

The two angles of a quadrilateral are known. All four angles sum to 360°.

2c + 110° + 120° = 360°

2c = 360° – 230°

2c = 130°

c = 65°

Example 4: In the diagram given below, the lines l₁ and l₂ are parallel, and line T is transversal. Find ∠GDE.

Find Missing Angles

Solution:

Step 1

In the above diagram, m∠GDE and m∠ADE are the angles on the straight line l₁.

So, we have,

m∠GDE + m∠ADE = 180°….. (1)

Step 2

In the above diagram, m∠ADE and m∠BEF are corresponding angles, and corresponding angles are always congruent.

So, we have,

m∠ADE = m∠BEF

Step 3

In (1), replace m∠ADE with m∠BEF

(1) ⇒ m∠GDE + m∠BEF = 180°

Step 4

From the diagram given above, we have m∠GDE = 4x and m∠BEF = 6x. So, replace m∠GDE with 4x and m∠BEF with 6x.

4x + 6x = 180°

Combine like terms.

10x = 180°

Divide both sides by 10.

10x/10 = 180°/10

Simplify.

x = 18°

Step 5

Plug x = 18° in m∠GDE = 4x

m∠GDE = 4 · 18°

m∠GDE = 72°

Example 5: The measures of two angles of a triangle are 60° 53′ 51″ and 51° 22′ 50″, respectively. The measure of the third angle in the radian is____.

(a) 1.15 radian

(b) 11 radian

(d) 11.5 radian

(e) None of the above

Solution:

The sum of two angles = (60° 53′ 51″) + (51° 22′ 50″)

=114° 16′ 41″

Third angle of triangle = 180° – (114° 16′ 41″) = 65° 43′ 19″

\(\begin{array}{l} =\left(65+\frac{43}{60}+\frac{19}{3600} \right)^\circ=\frac{234000+2580+19}{3600}\\=\left( \frac{236599}{3600} \right)^\circ=(65.722)^\circ\end{array} \)

We know that

\(\begin{array}{l}1^\circ=\frac{\pi }{180}\ \text{radian}\ (65.722){}^\circ =\left( \frac{\pi }{180}\times 65.722 \right) \text{radian} = 1.15\ \text{radian}\end{array} \)

Example 6: Find the ratio of radii of two circles, if the arc of equal length subtends angles 30° and 45° at their centre.

(a) 2:3

(b) 4:3

(d) 3:4

(e) None of the above

Solution:

Suppose radii of circles be r₁ and r₂ then

\(\begin{array}{l}{{\theta }_{1}}=\frac{l}{{{r}_{1}}} \text{and}\ {{\theta }_{2}}=\frac{l}{{{r}_{2}}}\end{array} \)

where

\(\begin{array}{l}{{\theta }_{1}}={{30}^{o}},{{\theta }_{2}}={{45}^{o}}\\ \frac{{{\theta }_{1}}}{{{\theta }_{2}}}=\frac{\frac{l}{{{r}_{1}}}}{\frac{l}{{{r}_{2}}}}\\ \Rightarrow \frac{{{\theta }_{1}}}{{{\theta }_{2}}}=\frac{{{r}_{2}}}{{{r}_{1}}} \Rightarrow \frac{{{r}_{1}}}{{{r}_{2}}}=\frac{{{\theta }_{2}}}{{{\theta }_{1}}}\Rightarrow \frac{{{r}_{1}}}{{{r}_{2}}}=\frac{45}{30} \Rightarrow \frac{{{r}_{1}}}{{{r}_{2}}}=\frac{3}{2}\\ {{r}_{1}}:{{r}_{2}}=3:2\\\end{array} \)

Example 7: The perimeter of a sector of a circle of area 64 π sq. cm is 56 cm, then the area of the sector is

(A) 140 sq.cm

(B) 150 sq.cm

(D) 170 sq.cm

Solution:

A = 64 π

∴ πr² = 64π

∴ r = 8 cm

Since, perimeter of sector = 2r + S

∴ S + r + r = 56

∴ S + 8 + 8 = 56

∴ S = 40

Since, S = rθ

∴ 40 = 8 × θ

∴ θ = 5c

∴ Area of sector

\(\begin{array}{l}=\frac{1}{2}r^{2} \theta = \frac{1}{2}[8^{2} \times 5]\end{array} \)

= 160 sq.cm.

Also read

Trigonometry

Systems of Measurement of Angles

Frequently Asked Questions

Define an angle.

We can define angle as the rotation from the initial point to an endpoint of a ray.

What is the measure of complete angle?

The measure of a complete angle is 360°.

Give the relation between radian and degree.

π radian = 180°.

Give the relation between radian, degree and grade measures.

If D is the number of degrees, R is the number of radians, and G is the number of grades in angle θ, then D/90 = G/100 = 2R/π.

Measurement of Angles Examples

How to Measure an Angle?

Measurement of Angles Formula

Relation between Degree, Radian and Grade Measure