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 \((r=\frac{a}{\sqrt{3}})\).

 

can't you get coordinate of midpoint on s? radius is distance from it to origin

 

Use the cosine formula.

 

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.

 

You have a triangle bounded by r, r and s.
The angle between r and r is 120°.
Use the cosine formula.

 

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.

 

Thanks to everyone, very clear now.

 

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.

 

Thanks to everyone, very clear now.

Also remember whenever you find a solution to your problem posted on AAC, do others a favor and say how you solved the probem.