Measuring Angles

What is an Angle in Geometry?

In two-dimensional geometry, an angle is a geometric figure formed when two rays meet at a common point. This common point is called the vertex, and the two rays are known as the sides (or arms) of the angle. An angle represents the amount of rotation or turning between the two rays and is a fundamental concept in geometry and trigonometry.

The term angle is derived from the Latin word “Angulus”, meaning corner, which accurately describes the shape formed by two intersecting rays.

Parts of an Angle

Every angle consists of three main components:

• Vertex: the common point where the two rays meet

• Initial side: the original or starting ray

• Terminal side: the final position of the ray after rotation

An angle is represented using the symbol $\angle$, and angle measures are often denoted using Greek letters such as $\theta$, $\alpha$, $\beta$, etc.

Positive and Negative Angles

Angles are classified based on the direction of rotation from the initial side to the terminal side.

Positive Angle

If an angle is measured in an anticlockwise direction, it is called a positive angle.
This is the standard convention used in trigonometry and coordinate geometry.

Negative Angle

If an angle is measured in a clockwise direction, it is called a negative angle.
Negative angles are commonly encountered in rotational motion and trigonometric graphs.

Common Terminology Used in Angles

The following terms are frequently used while studying angle measurement:

• Initial side: the ray from which the angle measurement begins

• Terminal side: the ray at which the angle measurement ends

• Vertex: the fixed point of rotation

These terms are especially important when working with standard position of angles, radians, and trigonometric functions.

Types of Angles Based on Measurement

In geometry, angles are classified into different types based on their degree measure:

• Acute Angle: an angle whose measure lies between $0^\circ$ and $90^\circ$

• Right Angle: an angle whose measure is exactly $90^\circ$

• Obtuse Angle: an angle whose measure lies between $90^\circ$ and $180^\circ$

• Straight Angle: an angle whose measure is exactly $180^\circ$

• Reflex Angle: an angle whose measure is greater than $180^\circ$ but less than $360^\circ$

• Complete Angle: an angle representing a full rotation, equal to $360^\circ$

Measurement of angle

In order to draw an angle, start by drawing two rays. Fix one ray in a place, and then rotate the other ray. The fixed ray is known as the initial side, while the rotated ray is the terminal side. And the measure of angle is amount of rotation from intitial to the terminal side.

An angle in standard position is if its vertex is located at the origin and initial side extends to positive $x$ axis. It can be seen in the figure below:

If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle.

If the angle is measured in a clockwise direction, the angle is said to be a negative angle.

How do we measure an Angle?

To measure an angle, there are basic mathematical tools such as protractor and compass. A protractor is used to provide the exact measure of angle and compass helps in constructing an angle. An angle can be measured in three ways namely as degrees, radians and revolution.

Degrees

Degree is unit of measure of angle. It is measured using a tool known as protractor. It is denoted by the symbol, '°'. A circle revolves completely at 360°, as the circle divides itself into 360 equal parts.

Radians

Radian is a measurement unit of angle, is used as an alternative of degrees. Radian is the ratio of length of arc which the angle subtends on a circle, divided by length of radius of the circle. Radian is an angle subtended by the arc of the length of the radius of the same circle at the center and the ratio will give the radian measure of the angle. Radian is denoted as rad.

Revolution

A circle completes its full revolution and has 360° as subdivision of circle. It refers to some full rotation of 360 degrees.

Relation between Degree and Radian Measures

In geometry and trigonometry, angles can be measured using different units, the most common being degrees and radians. To clearly understand angle measurement, it is important to know the relationship between degree measure and radian measure.

By definition, the angle subtended by a complete circle at its centre is measured as:

• $360^\circ$ in the degree system

• $2\pi$ radians in the radian system

Since both represent the same full rotation, we equate them as
$2\pi \text{ radians} = 360^\circ$

Dividing both sides by $2$, we get
$\pi \text{ radians} = 180^\circ$

Conversion between Degree and Radian

From the basic relation $\pi \text{ radians} = 180^\circ$, we obtain the standard conversion formulas:

• $1 \text{ radian} = \dfrac{180^\circ}{\pi} \approx 57^\circ 16'$

• $1^\circ = \dfrac{\pi}{180} \text{ radian} \approx 0.01746 \text{ radian}$

These formulas are widely used in trigonometry, calculus, and physics.

Degree to Radian Conversion Table

The following table shows commonly used angles converted from degrees to radians:

This table is extremely useful for exam preparation and trigonometric calculations.

Systems Used for the Measurement of Angles

There are three main systems used for measuring angles in mathematics:

1. Sexagesimal System

2. Centesimal System

3. Circular System

Sexagesimal System of Angle Measurement

In the sexagesimal system, angles are measured in degrees, minutes, and seconds.

• $1$ right angle $= 90^\circ$

• $1^\circ = 60'$ (minutes)

• $1' = 60''$ (seconds)

This system is commonly used in basic geometry, navigation, and surveying.

Centesimal System of Angle Measurement

In the centesimal system, a right angle is divided into 100 equal parts, each part called a grade.

• $1$ right angle $= 100g$

This system is less common but is sometimes used in engineering and technical applications.

Circular System of Angle Measurement (Radian Measure)

In the circular system, angles are measured in radians.

Definition of One Radian

One radian is defined as the measure of a central angle of a circle that subtends an arc whose length is equal to the radius of the circle.

A central angle is an angle formed at the centre of a circle by two radii.

Formula for Radian Measure

If an arc of length $l$ subtends an angle at the centre of a circle of radius $r$, then the radian measure of the angle is:

$\text{Radian measure} = \dfrac{l}{r}$

Since the total circumference of a circle is $2\pi r$, a full circular rotation corresponds to:

$\dfrac{2\pi r}{r} = 2\pi \text{ radians}$

Thus,

• $2\pi \text{ radians} = 360^\circ$

• $\pi \text{ radians} = 180^\circ$

• $1 \text{ radian} = \dfrac{180^\circ}{\pi} \approx 57.3^\circ$

Interconversion of Units of Angle Measurement

In geometry and trigonometry, angles can be measured in different units such as degrees, radians, and grades. Since these units describe the same physical quantity (rotation), it is essential to know how to convert one unit into another. This process is called interconversion of angle units and is widely used in trigonometry, calculus, physics, and engineering applications.

Degree to Radian Conversion

Since both degrees and radians measure angles, we can convert between them using a simple proportion.

If $\theta$ represents the angle in degrees and $\theta_R$ represents the angle in radians, then the relation is:

$\dfrac{\theta}{180} = \dfrac{\theta_R}{\pi}$

This can also be written as:

$\dfrac{\text{Degree}}{180} = \dfrac{\text{Radian}}{\pi}$

Using this relation, any angle measured in degrees can be converted into radians easily.

Important Notes on Degree–Radian Conversion

• Radian is a unit of angle measurement, not a number.

• $\pi$ is a real number (approximately $3.1416$); it does not represent $180^\circ$.

• The correct relation to remember is $\pi \text{ radians} = 180^\circ$.

Arc Length Formula in Radian Measure

In a circle of radius $r$, if an angle $\theta$ (in radians) subtends an arc of length $s$, then:

$\text{Arc length } s = r\theta$

This formula is valid only when the angle is measured in radians and is extremely important in circular motion and calculus.

Degree to Grade Conversion

In the centesimal system, angles are measured in grades.

Let:

• $D$ = number of degrees

• $G$ = number of grades

The relation between degrees and grades is:

$\dfrac{D}{90} = \dfrac{G}{100}$

This relation helps convert degree measure into grade measure and vice versa.

Radian to Grade Conversion

Let:

• $R$ = number of radians

• $G$ = number of grades

The relation between radians and grades is given by:

$\dfrac{G}{100} = \dfrac{2R}{\pi}$

This formula is useful when converting angles between the circular system and the centesimal system.

Measurement of Angles Using a Protractor

An angle is commonly measured using geometric instruments such as a protractor and a compass. While a compass is mainly used for constructing angles, a protractor is used for both constructing and measuring angles.

A protractor is one of the most important tools in geometry because it allows accurate angle measurement in degrees and helps visualize angle size.

Steps to Measure an Angle Using a Protractor

1. Place the centre of the protractor exactly on the vertex of the angle.

2. Align one arm of the angle with the zero line of the protractor.

3. Read the value on the scale where the other arm of the angle meets the protractor.

4. The reading obtained gives the measure of the angle in degrees.

Constructing Angles Using a Protractor

A protractor can also be used to construct angles of a given measure, which is important for geometric constructions and accuracy practice.

Steps to Construct an Angle Using a Protractor

1. Draw a straight baseline using a ruler.

2. Mark a point $O$ on the line and place the centre of the protractor at $O$.

3. Align the baseline with the zero line of the protractor.

4. Locate the required angle measure on the inner scale and mark it as point $C$.

5. Using a ruler, join points $O$ and $C$ to form the required angle.

Solved Examples Based on Measurement of Angles

Example 1: What is the shortest positive measure of the angle (in degrees) formed between the positive x-axis and the line if the total angle elapsed is $810^\circ$?

Solution:
Total angle elapsed $= 810^\circ$
One complete revolution $= 360^\circ$
Two complete revolutions $= 720^\circ$

Thus, the required angle $= 810^\circ - 720^\circ = 90^\circ$

Hence, the answer is $90^\circ$.

Example 2: What is the value of the radius of a circle if the circumference is $\dfrac{8\pi}{3}$?

Solution:
Circumference of a circle $= 2\pi r$

So, $2\pi r = \dfrac{8\pi}{3}$

$\Rightarrow r = \dfrac{4}{3}$

Hence, the answer is $\dfrac{4}{3}$.

Example 3: The angles of a triangle are in the ratio $2:3:5$. Find the least angle in radians.

Solution:
Let the angles be $2x$, $3x$, and $5x$.

Sum of angles of a triangle $= 180^\circ$

So, $2x + 3x + 5x = 10x = 180^\circ$

$\Rightarrow x = 18^\circ$

Converting to radians,
$x = 18^\circ = 18 \times \dfrac{\pi}{180} = \dfrac{\pi}{10}$

Minimum angle $= 2x = 2 \times \dfrac{\pi}{10} = \dfrac{\pi}{5}$ radians.

Hence, the answer is $\dfrac{\pi}{5}$ radians.

Example 4: If the radius of a circle is half the arc length subtended by an angle $A$, find the measure of angle $A$.

Solution:
Given $r = \dfrac{l}{2}$

So, $\dfrac{l}{r} = 2$

But angle in radians $= \dfrac{l}{r}$

Therefore, $A = 2$ radians.

Hence, the answer is $2$ radians.

Example 5: A circular wire of radius $3$ cm is cut and bent so as to lie along a circle of radius $48$ cm. Find the angle subtended by the wire at the centre of the circle.

Solution:
Length of the circular wire $= 2\pi r = 2\pi \times 3 = 6\pi$ cm

Angle subtended at the centre $= \dfrac{\text{arc length}}{\text{radius}} = \dfrac{6\pi}{48} = \dfrac{\pi}{8}$ radians

Hence, the answer is $\dfrac{\pi}{8}$ radians.

List of Topics Related to Measuring Angles

This section presents a comprehensive list of topics related to measuring angles, covering all key concepts. It helps learners quickly identify connected topics for systematic study and effective revision.

NCERT Resources

This section offers important NCERT-based resources for Class 11 Chapter 3 – Trigonometric Functions, aimed at building strong conceptual clarity. It includes concise notes, step-by-step NCERT solutions, and NCERT Exemplar problems to help you confidently understand and apply trigonometric ratios.

NCERT Class 11 Chapter 3 Trigonometric Functions Notes

NCERT Class 11 Chapter 3 Trigonometric Functions Solutions

NCERT Exemplar Class 11 Chapter 3 Trigonometric Functions

Practice Questions based on Measuring Angles

This section includes practice questions based on measuring angles to help you apply the concepts learned, such as degree and radian conversion, types of angles, arc length, and protractor-based measurement. These questions are designed to strengthen problem-solving skills and improve accuracy for exams and real-world applications.

Measuring Angles- Practice Question MCQ

Below is the list of the next topics which you can practice: