Thread: find area of infinite circles

Here is a neat little problem involving infinite series I thought was cool.

Please excuse my haphazard 'paint' drawing.

If the circles go off into the vertices to infinity, find the area of the infinite circles inscribed in the equilateral triangle with sides length s.

Hi.

I have used : The inradius of an equilateral triangle has radius r = a/3
where a is the altitude

After making a new tangent to the incircle parralel to the AB,
we get a new triangle similar to the earlier triangle
( ie; an equilateral triangle with side S'= S/3)

the incircle of this triangle is our next required circle
and thus we get its radius(r) as a'/3










Hence the new radius becomes







Similarly


Thus the sum now becomes


This in an infinite GP with first term



And common difference


Thus the sum of terms is given by



We needed to get it for three sides so sum is

Cool, Adarsh. I got the same thing. I was wondering if I was right or not.

Since we agree, I suppose I was.

These types of problems are always fun.