Question

a regular hexagon is inscribed in a circle with a radius of 10 cm. find the area of the shaded region shown. (use the exact answer. use pi as \\(\pi\\) in any solution for.)

Explanation:

Step1: Find the area of the circle

The formula for the area of a circle is $A_{circle}=\pi r^{2}$, where $r = 10$ cm. So $A_{circle}=\pi\times10^{2}=100\pi$ $cm^{2}$.

Step2: Find the area of the regular hexagon

A regular hexagon can be divided into six equilateral triangles. The side - length of the regular hexagon inscribed in a circle of radius $r$ is equal to the radius of the circle, i.e., $a = r=10$ cm. The area of an equilateral triangle with side - length $a$ is $A_{\triangle}=\frac{\sqrt{3}}{4}a^{2}$. For $a = 10$ cm, $A_{\triangle}=\frac{\sqrt{3}}{4}\times10^{2}=25\sqrt{3}$ $cm^{2}$. The area of the hexagon $A_{hexagon}=6\times A_{\triangle}=6\times25\sqrt{3}=150\sqrt{3}$ $cm^{2}$.

Step3: Find the area of the shaded region

The area of the shaded region $A = A_{circle}-A_{hexagon}=100\pi - 150\sqrt{3}$ $cm^{2}$. Calculate the numerical value: $A\approx100\times3.14159-150\times1.73205=314.159 - 259.8075 = 54.3515\approx54.4$ $cm^{2}$.

Answer:

$54.4$ $cm^{2}$