Table of Contents

Last modified on August 3rd, 2023

A central angle is an angle formed between two different radii of a circle. They are angle subtended to the center of a circle from two different points. Thus the vertex of the central angle will always be the center point of a circle.

The above figure shows a circle with center O having a central angle, ∠AOB. The two arms forming the angle are OA and OB where the two points A and B can be isolated points or they could be the end points of an arc and chord.

**Prove Central Angle Theorem**

To prove:

∠AOB = 2∠ACB

Proof:

Let ∠AOB be the central angle, and ∠ACB the inscribed angle of the circle.

Since lines OC, OA, and OB are the radius of the circle and are of the same length,

Hence,

△COA and △COB are both isosceles triangles.

Let, ∠ACO = α_{1} and ∠BCO = α_{2}

Then we can write,

∠CAO = α_{1 }and ∠CBO = α_{2} (Isosceles triangles)

Again,

∠COA = 180° – 2α_{1}….. (1)(Sum of the angles in a triangle is 180°)

Similarly,

∠COB = 180° – 2α_{2}….. (2)

Since, all angles at center O add up to 360°,

∠AOB + ∠COA + ∠COB = 360°

∠AOB = 360° – (∠COA + ∠COB)…….. (3)

Substituting the value of (1) and (2) in (3) we get,

∠AOB = 360° -(180° – 2α_{1}) – (180° – 2α_{2})

∠AOB = 360° – 180° + 2α_{1 }– 180° + 2α_{2}

∠AOB = 2α_{1 }+ 2α_{2}

Thus,

∠AOB =2∠ACB**Hence Proved**

Central angle of a circle can be determined if its corresponding inscribed angle is known by using the formula derived from the central angle theorem given below:

Central angle = 2 x Inscribed angle

**Alternative formula**:

The central angle of a circle can also be determined if the arc length and the radius forming the angle is know, using the following formula:

Central angle = Arc length x 360°/2πr

Let us solve some problems to understand the concepts better.

**Solve the missing angle x in the diagram given below.**

Solution:

As we know,

According to the central angle theorem the measure of the central angle in twice the measure of the central angle,

Given,

Inscribed angle = 80°

Thus,

Inscribed angle = 2 x 80° =160°

**Find the central angle, where the arc length is 30 cm and the length of the radius is 15 cm. Given π = 3.14.**

Solution:

As we know,

Central angle = Arc length x 360°/2πr, here arc length = 30 cm, r = 15 cm

= 30 x 360°/2 x 3.14 x 15

= 10800°/94.2

= 114.64°

**If the central angle of a circle is 88**° **and the arc length formed is 28 cm. Find the radius of the circle. Given π = 3.14.**

Solution:

As we know,

Central angle = Arc length x 360°/2πr, here central angle = 88°arc length = 28 cm

88° = 28 x 360°/ 2 x 3.14 x r

r = 28 x 360°/2 x 3.14 x 88

= 10080°/552.64

= 18.23 cm

Last modified on August 3rd, 2023