Question

Find the ratio of the areas of the in-circle and circum-circle of an equilateral triangle?

Please give the answer in details.

Answer 1

`1:4`

Explanation:

Given `Delta ABC =` equilateral triangle
Let radius of in-circle be `r`, and radius of circumcircle be `R`.
In `Delta OBD, angleOBD=30^@, angle ODB=90^@ => R=2r`

Let area of in-circle be `A_I` and area of circumcircle be `A_C`,

`=> A_I/A_C=(pir^2)/(pi(R^2))=(pir^2)/(pi(2r)^2)=1/4`

Hence, `A_I : A_C = 1:4`