Thread: [SOLVED] Triangles and circles

I'm having a problem with a question that involves a circle contained within an equilateral triangle which is also contained by a bigger circle. No sides are given; the question asks to find the ratio between the areas of the two circles. Is there a simple relationship between the two...

Hello orangeiv

Originally Posted by orangeiv

I'm having a problem with a question that involves a circle contained within an equilateral triangle which is also contained by a bigger circle. No sides are given; the question asks to find the ratio between the areas of the two circles. Is there a simple relationship between the two...

Yes - the radius of the circumcircle is twice the radius of the incircle, and its area is therefore 4 times as great. Here's why:

• Draw a diagram showing an equilateral triangle with its incircle and circumcircle - the two circles you refer to in your question.

• Draw the three medians of the triangle. They meet at a point, G say, which divides the line joining each vertex of the triangle to the mid-point of the opposite side in the ratio 2:1.

G is the centre of both circles, and the ratio of their radii is therefore this same ratio 2:1.

Grandad