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Central Angle of a Circle Formula

The angle between two radii of a circle is known as the central angle of the circle. The two points of the circle, where the radii intersects in the circle (Note – The other end of the radii meets at the center of the circle), forms a segment of the Circle called the Arc Length.
Central Angle of a Circle Formula
Formula to Find the Central Angle of a Circle – 
\[\LARGE Central\;Angle\;\theta =\frac{Arc\;Length\times 360}{2\pi r}\]

Solved Examples

Example 1: Find the central angle, where the arc length measurement is about 20 cm and the length of the radius measures 10 cm?


Given r = 10 cm

Arc length = 20 cm

The formula of central angle is,

Central Angle $\theta$ = $\frac{Arc Length \times 360}{2 \times\pi \times r}$

Central Angle $\theta$ = $\frac{20 \times 360}{2 \times 3.14 \times 10}$

Central Angle $\theta$ = $\frac{7200}{62.8}$ = 114.64°

Example 2: If the central angle of a circle is 82.4° and the arc length formed is 23 cm then find out the radius of the circle.


Given Arc length = 23 cm

The formula of central angle is,

Central Angle $\theta$ = $\frac{Arc\;Length \times 360}{2\times\pi \times r}$

82.4° = $\frac{23 \times 360}{2\times\pi \times r}$

82.4° = $\frac{8280}{6.28\times r}$

r= $\frac{8280}{6.28\times 82.4}$

r = 16 cm

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