Question

A circle is inscribed in an equilateral triangle. If the diameter of the circle is 2, what is the area of the region inside the triangle not including the area of the circle?

Answer 1

If s is a side of triangle

s= 6/sqrt 3

Triangle Area is sqrt 3/4 * s^2 = 3 sqrt 3

So required area is 3 sqrt 3 - pi

Answer 2

Why does s = 6/ sqrt 3 ?

Answer 3

Just split the triangle down the middle and use Pythagorus and the 60 degree angle to solve the triangle and everything will pop out.

(It's actually 6/sqrt 3 * r but r=1)

Answer 4

Ok I get that the hypotenuse of the split triangle is 2x and the legs are x and x sqrt 3 but how do I solve for x?

Answer 5

OK, I meant use the triangle containing r (construct from triangle corner to circle center).

U have Tan 30 =1/ sqrt 3 = r/ s/2 = 2r/s so r =1 = s/2sqrt3 = s sqrt 3/6

Answer 6

Ohh ok i get it now tysm :)