Q. 17 - Precalculus Enhanced with Graphing Utilities

Question

An equilateral triangle is inscribed in a circle

of radius r. See the figure in Problem 16. Express the area A within the circle, but outside the triangle, as a function of the length x of a side of the triangle.

Step-by-Step Solution

Answer

The area within the circle, but outside the triangle, as a function of the length x of a side of the triangle is $(\frac{\mathrm{\pi }}{3}-\frac{\sqrt{3}}{4})x^2$.

Step 1. Find the area of the circle using r 2 = x 2 3 .

Area of the whole circle $$\begin{gathered} ={\mathrm{\pi r}}^2 \\ =\mathrm{\pi }\left(\frac{{\mathrm{x}}^2}{3}\right) \end{gathered}$$

Area of the equilateral triangle with length of the sides, x $=\frac{\sqrt{3}}{4}{x}^2$.

Step 2. Subtract the area of equilateral triangle from the the area of the circle to find the required area.

So the area A within the circle, but outside the triangle, as a function of the length x of the side of the triangle is,

$$\begin{gathered} A=\frac{{\mathrm{\pi x}}^2}{3}-\frac{\sqrt{3}}{4}{x}^2 \\ A=(\frac{\mathrm{\pi }}{3}-\frac{\sqrt{3}}{4})x^2 \end{gathered}$$