Area of an equilateral triangle if the radius of....?

Question

Hey I'm having difficulty with this math problem:

Find the area of an equilateral triangle if the radius of its inscribed circle is 3.

Could someone enlighten me as to what steps I should take to obtain the answer?

Answer 1

Try drawing lines connecting the circumference to the centre of the circle. A number of new shapes and angles pop up. Knowing that the radius is 3, the solution quickly follows.

Answer 2

1. Draw a picture of the equilateral triangle with the inscribe circle.
2. Draw a line from one vertex through the center of the circle and extend it to the side opposite the vertex.
3. You should note that this is the altitude of the triangle.
4. Note that the length of the altitude = the distance from the vertex to the center of the circle (call this distance x) + the radius of the circle
5. Now draw a line fro the center of the circle to to the point of tangency on a different side.
6. You should see that a right triangle is formed that has hypotenuse x and one leg = r and the other = s/2 where s i the length of a side of the equilateral triangle
7. Use he pythagoren theorem to get x= sqrt(s^2/4 + r^2)
8. So altitude = h = r +x = r+ sqrt(s^2/4+r^2)
9. So Area = sh/2 = (s/2)(r+sqrt(s^2/4 +r^2)