Measuring Angles

An angle is formed when two rays meet at a common endpoint. This endpoint is called the vertex, and the two rays are called the arms of the angle. The amount of rotation or opening between the arms is called the measure of the angle, which is usually expressed in degrees or radians.

Measure of an angle

An angle is measured using a protractor, which is a semi-circular tool marked from 0° to 180°.

• The outer scale measures angles in the clockwise direction from 0° to 180°.
• The inner scale measures angles in the anticlockwise direction from 0° to 180°.

Angles are commonly measured in three units: degrees, radians, and revolutions.

• Degrees (°): A full circle is divided into 360°, and a semicircle measures 180°. One complete rotation is equal to 360°.
• Radians (rad): This is the SI unit of angle measurement. It is defined as the ratio of the arc length to the radius of a circle. One complete rotation equals 2π radians, so π radians equals 180°.
• Revolution: It represents one complete turn of a circle. One revolution is equal to 360° or 2π radians.

Degrees and Radians Conversion

A circle subtends 2π radians or 360° at its centre. So, 2π radians = 360°

From this, we get:

• π radians = 180°
• 1 radian = 180° / π ≈ 57.3°
• 1° = π / 180 ≈ 0.0174 radians.

Formulas

• Angle in Radian = Angle in Degree × π/180
• Angle in Degree = Angle in Radian × 180/π

Measuring Angles Using a Protractor

We use a protractor to measure angles. Consider ∠AOB shown below. From its opening, it appears to be an acute angle, meaning its measure lies between 0° and 90°.

Steps to Measure the Angle

Step 1: Place the protractor such that its centre coincides with point O and align the baseline with ray OB. Start reading from the 0° mark on the inner scale.

Step 2: Observe where the second ray OA intersects the scale of the protractor. The reading at this point gives the measure of the angle.

Thus, the measure of ∠AOB = 37°, which confirms that it is an acute angle.

How to Measure an Obtuse Angle

Consider ∠the AOC shown below. From its opening, it is clear that the angle is greater than 90° and less than 180°, so it is an obtuse angle.

Steps to Measure the Angle

Step 1: Place the protractor such that its centre coincides with point O and align the baseline with ray OC. Start reading from the 0° mark on the outer scale (bottom-left side).

Step 2: Observe where the ray OA intersects the outer scale of the protractor. The reading at this point gives the measure of the angle.

Thus, the measure of ∠AOC = 143°, which confirms that it is an obtuse angle.

Solved Examples

Example 1: Measure the angle ∠ABC using the protractor.

Solution:

We can easily measure the angle using the protector as shown in the image below,

👁 protractor_4

Example 2: In triangle ABC, use the protractor and measure ∠CAB.

Solution:

The measure of the angle ∠CAB of triangle ABC is found using the protector.

👁 protractor_5

Example 3: Convert 90 degrees to radians.

Solution: 

Given, angle 90°

We know that,

Angle in Radian = Angle in Degree × (π/180)

⇒ 90° = 90 × (π/180)
⇒ 90° = π/2

Hence, 90 ° is equal to π/2 radian.

Example 4: Convert π/6 rad into degrees.

Solution: 

Given, angle π/6 rad

We know that,

Angle in Degree = Angle in Radian × (180/π)

⇒ π/6 rad =  π/6 × (180/π) 

⇒ π/6 rad = 180/6°

⇒ π/6 rad = 30°

Related Articles

• Types of Angles
• Angles Formula