Question

The central angle of a sector is ${{72}^{\circ }}$ and the sector has an area of $5\pi$. How do you find the radius?

Answer 1

Hint: To solve these questions first convert the angle given in degrees to radians by using the formula for the conversion. Then substitute the value of the angle in radians in the formula to calculate the area of a sector. After substituting the values, simplify the expression to get the final answer.

Complete step by step solution:
Given:
The angle of the sector ${{72}^{\circ }}$. Convert the angle from degrees to radians by using the formula ${{1}^{\circ }}=\dfrac{\pi }{{{180}^{\circ }}}$.
Since, ${{180}^{\circ }}=\pi radians$
Therefore, ${{72}^{\circ }}$in radians will be ${{72}^{\circ }}\times \dfrac{\pi }{{{180}^{\circ }}}=\dfrac{2\pi }{5}$
Now, area of a sector can be calculated by using the formula
$A=\dfrac{1}{2}{{r}^{2}}\theta$, where $r$ is the radius of the circle or the sector that is given, $\theta$is measured in radians and $A$ is the area of the sector.
Substituting the values of $A$ and $\theta$ given in the question in the formula, we get
$\Rightarrow 5\pi =\dfrac{1}{2}{{r}^{2}}\left( \dfrac{2\pi }{5} \right)$
Now, transposing all the constant values in the above expression to one side of the equation, we get
$\Rightarrow \dfrac{5\pi \times 2\times 5}{2\pi }={{r}^{2}}$
Cancelling the like terms from the numerator and denominator of the fraction, we get,
$\Rightarrow {{r}^{2}}=5\times 5$
By multiplying the remaining terms in the numerator, we get,
$\Rightarrow {{r}^{2}}=25$
Now, taking square roots on both the side of the equation we get,
$\Rightarrow \sqrt{{{r}^{2}}}=\sqrt{25}$
$\Rightarrow r=5$
Hence, the radius of a sector with a central angle of ${{72}^{\circ }}$ and the sector area as $5\pi$ is $r=5$ units.

Note: A sector can be defined as an area of circle enclosed between its radii and the arc joining the two radii. Sectors can be of two types: major sector and minor sector. In the above given question, while taking the square root of the radius, we will only consider the positive value of the radius, since radius cannot be negative ever as it is a measure of length.