Chapter 12, Problem 4PST

To determine

To calculate : The area of the hexagon inscribed in a circle.

Answer to Problem 4PST

The area of the hexagon inscribed in a circle will be $93.53c{m}^2$

Explanation of Solution

Given information : A hexagon is inscribed in a circle with a diameter of $12cm$ .

Formula used : When a hexagon is inscribed in the circle it makes 6 equilateral triangles each having side length equal to the radius of the circle. So, the area of the hexagon will be 6 times the area of the equilateral triangle. And we know the radius $(r)$ of a circle is half the diameter of the circle. And the area of the equilateral triangle will be $\frac{\sqrt{3}}{4}{r}^2$ ,

Calculation : Calculate the radius of the circle.

The radius will be $\frac{12}{2}=6cm$

And the area of the equilateral triangle will be $\frac{\sqrt{3}}{4}{r}^2$

  $\begin{array}{l}=\frac{\sqrt{3}}{4}(6^2)\\ =\frac{\sqrt{3}}{4}\times 36\\ =9\sqrt{3}c{m}^2\end{array}$

Now, calculate the area of the hexagon.

That will be 6 times the area of the equilateral triangle.

  $\begin{array}{l}=6\times 9\sqrt{3}c{m}^2\\ =54\sqrt{3}c{m}^2\\ =93.53c{m}^2\end{array}$