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 intersect in the circle (Note – The other end of the radii meets at the centre of the circle), forms a segment of the Circle called the Arc Length.
Formula to Find the Central Angle of a Circle –
\[ Central\;Angle\;\theta =\frac{Arc\;Length\times 360^{o}}{2\pi r}\]
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.
Solution:
Given,
r = 10 cm
Arc length = 20 cm
The formula of central angle is,
Central Angle
\(\begin{array}{l}\theta\end{array} \)
= \(\begin{array}{l}\frac{Arc Length \times 360^{o}}{2 \times\pi \times r}\end{array} \)
Central Angle
\(\begin{array}{l}\theta\end{array} \)
= \(\begin{array}{l}\frac{20 \times 360^{o}}{2 \times 3.14 \times 10}\end{array} \)
Central Angle
\(\begin{array}{l}\theta\end{array} \)
= \(\begin{array}{l}\frac{7200}{62.8}\end{array} \)
= 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.
Solution:
Given,
Arc length = 23 cm
The formula of central angle is,
Central Angle
\(\begin{array}{l}\theta\end{array} \)
= \(\begin{array}{l}\frac{Arc\;Length \times 360^{o}}{2\times\pi \times r}\end{array} \)
82.4° =
\(\begin{array}{l}\frac{23 \times 360^{o}}{2\times\pi \times r}\end{array} \)
82.4° =
\(\begin{array}{l}\frac{8280}{6.28\times r}\end{array} \)
r=
\(\begin{array}{l}\frac{8280}{6.28\times 82.4}\end{array} \)
r = 16 cm
Very nice answer and solution.