An equilateral triangle is inscribed in a circle of radius r. Show that the ratio of the area of the triangle Class 9

An equilateral triangle is inscribed in a circle of radius r. Show that the ratio of the area of the triangle Class 9

Question 1.

An equilateral triangle is inscribed in a circle of radius r. Show that the ratio of the area of the triangle to the area of the circle is equal to $\frac{3 \sqrt{3}}{4 \pi} \approx 0.413$. Class 9

Solution:

∵ An equilateral triangle is inscribed in a circle of radius r.

∴ Area of triangle = $\frac{a b c}{4 r}$ ... [Where r is circum radius]

$\therefore \quad \frac{\sqrt{3}}{4} \times(\text { side })^2=\frac{(\text { side })^3}{4 r}$

∵ Side of equilateral triangle = $\sqrt{3} r$

∴ Area of equilateral triangle

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

Also, area of circle = nπ²

Required ratio

$\begin{aligned} & =\frac{\text { Area of triangle }}{\text { Area of circle }} \\ & =\frac{\frac{3 \sqrt{3}}{4} r^2}{\pi \mathrm{r}^2}=\frac{3 \sqrt{3}}{4 \pi}\end{aligned}$

∴ Ratio = $\frac{3 \sqrt{3}}{4 \pi} \approx 0.413$