# Geometry: Finding the radius of circumcircle

Discussion in 'Homework Help' started by Agonche, Jun 3, 2013.

1. ### Agonche Thread Starter Member

Aug 26, 2011
30
0
I need help finding the radius of equilateral triangle circumscribed circle.

I found the height of the triangle which is: $h=\frac{\sqrt{3}}{2}a$
Using Heron's formula or the simple formula $A=\frac{bh}{2}$ I found the area of the equilateral triangle: $A=\frac{a^{2}\sqrt{3}}{4}$

But I need to find the circumference of the circumscribed circle, and to do that I need to know the radius of the circle.
I found the solution but I need to know how to come to that solution $(r=\frac{a}{\sqrt{3}})$.

2. ### panic mode Senior Member

Oct 10, 2011
1,657
468
can't you get coordinate of midpoint on s? radius is distance from it to origin

3. ### MrChips Moderator

Oct 2, 2009
18,198
5,717
Use the cosine formula.

4. ### WBahn Moderator

Mar 31, 2012
23,862
7,384
Focus on the following small triangle:

The sides formed by r (the quantity you are looking for), the right one-half of the bottom side of the equilateral triangle, and the vertical portion of the height from the center of the circuit down to the bottom side?

If you are able to use trig, then you know that the angle on the right-hand corner of this triangle is 30°. You are now two lines of algebra away from the solution.

If you want to use just geometry, then write the three sides of the triangle in terms of s, h, and r. You are now three lines of algebra away from the solution. You already know h in terms of s, so now you just have three sides in terms of s and r. Use the Pythagorean theorem to solve for r.

5. ### MrChips Moderator

Oct 2, 2009
18,198
5,717
You have a triangle bounded by r, r and s.
The angle between r and r is 120°.
Use the cosine formula.

6. ### WBahn Moderator

Mar 31, 2012
23,862
7,384
He may not be allowed to use any trig. If so, then he can't use the cosine formula. But it isn't needed. It can be solved in three lines with the Pythagorean theorem and there isn't even any need to solve a quadratic equation because the r^2 terms cancel.

7. ### Agonche Thread Starter Member

Aug 26, 2011
30
0
Thanks to everyone, very clear now.

8. ### WBahn Moderator

Mar 31, 2012
23,862
7,384
Which way did you solve it? I'd recommend solving it both ways (plus any other way that comes to mind) to get practice with different ways of approaching the same problem.

9. ### MrChips Moderator

Oct 2, 2009
18,198
5,717
Also remember whenever you find a solution to your problem posted on AAC, do others a favor and say how you solved the probem.

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