A hexagon is inscribed in a circle of radius r. Show that the ratio of the area of the hexagon to the area Class 9

A hexagon is inscribed in a circle of radius r. Show that the ratio of the area of the hexagon to the area Class 9

Question 1.

A hexagon is inscribed in a circle of radius r. Show that the ratio of the area of the hexagon to the area of the circle is equal to $\frac{3 \sqrt{3}}{2 \pi} \approx 0.827$. Class 9

Solution:

∵ A regular hexagon is inscribed in a circle of radius r

If all the vertices are joined with centre of circle then six equilateral triangles are formed each with side r units.

∴ Each side of hexagon subtends 60° at the centre

∴ Area of one equilateral triangle = $\frac{\sqrt{3}}{4} r^2$

Area of hexagon

= 6 × area of one equilateral triangle

$\begin{aligned} & =\frac{6 \sqrt{3}}{4} r^2 \\ & =\frac{3 \sqrt{3}}{2} r^2\end{aligned}$

∵ Area of circle = πr²

∴ Required ratio

$\begin{aligned} & =\frac{\text { Area of hexagon }}{\text { Area of circle }} \\ & =\frac{\frac{3 \sqrt{3}}{2} r^2}{\pi r^2} \\ & =\frac{3 \sqrt{3}}{2 \pi} \\ & \approx 0.827\end{aligned}$

∴ Ratio of the area of the hexagon to the area of the circle = $\frac{3 \sqrt{3}}{2 \pi}$ = 0.827

$\because \quad \frac{3 \sqrt{3}}{2 \pi}=2 \times \frac{3 \sqrt{3}}{4 \pi}$

∴ It is twice the ratio for equilateral triangle to the area of circle.