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Central Angle

The central angle of a circle is formed by two radii. The circle consists of arcs and central angles. The central angle is measured with the help of the arc length and the radius. The central angle problems can be solved with the help of the central angle formula.

Central Angle
In the above figure, the central angle of the circle is the angle $\theta$  at the central part of the circle formed between two radii ON and OM. Let us go ahead and learn more about the central angle of a circle.

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Central Angle Formula

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The formula for central angle of a circle is given as follows,

Central angle, $\theta$ = $\frac{A \times 360}{2 \times \pi \times r}$

Where, A represents for the arc length of the circle r denotes the radius of circle and $\pi$ denotes the constant value 3.14 or $\frac{22}{7}$. We can find central angle of a circle with the help of this formula.

Examples

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Below are some examples based on central angle


Example 1: 
Find the central angle, where the radius length measures about 11 cm and the arc length measurement is about 14 cm?

Solution:

Now we find the central angle of the circle by using the formula,

$\theta$ = $\frac{Arc\ length\ \times 360}{2 \times \pi \times r}$

$\theta$ = $\frac{14\ \times\ 360}{2\ \times\ 3.14\ \times\ 11}$

Now we simplify,

we get, 14 $\times$ 360 = 5040

2 $\times$ 3.14 $\times$ 11 = 69.08

$\theta$$\frac{5040}{69.08}$

$\theta$ = 72.95

Thus, the central angle measures about 72.95 degrees

Example 2: Find the central angle, where the radius length measures about 12cm and the arc length measurement is about 15cm?

Solution:

Now, we can find the central angle of the circle by using the formula,

$\theta$ = $\frac{Arc\ length\ \times 360}{2 \times \pi \times r}$

$\theta$$\frac{15\ \times\ 360}{2\ \times\ 3.14\ \times\ 12}$

Now we simplify,

we get, 15 $\times$ 360 = 5400

2 $\times$ 3.14 $\times$ 12 = 75.36

$\theta$ = $\frac{5400}{75.36}$

$\theta$ = 71.66

Thus, the central angle measures about 71.66 degrees.

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