Chapter 11.5, Problem 30WE

To determine

To determine the areas of the two circles we got.

Answer to Problem 30WE

the ratio of the areas of these two circles is,

  $1:4$

Explanation of Solution

Given information :

Two circles are inscrobed and circumscribed to eqilatoral triangle.

To determine the ration of the areas of two circles, we shall determine the radius of the two circles.

Let a be the length of the equilatoral triangle

Radius of inscribed circle shall be $\frac{a}{\sqrt{3}}$ .

Radius of circumscribed circle shall be $\frac{2}{\sqrt{3}}a$

Hence the ratio of the areas of these two circles is,

  $\pi {\left(\frac{a}{\sqrt{3}}\right)}^2:\pi {\left(\frac{2a}{\sqrt{3}}\right)}^2=1:4$