# Thread: Diameter of circle--equilateral triangle

1. ## Diameter of circle--equilateral triangle

If I have an equilateral triangle with sides of 4", how do I figure out what the diameter of a circle would be that touches the three points?

2. ## Re: Diameter of circle--equilateral triangle

Draw it out; make sure the diagram isn't too small. Write EVERYTHING you know about the circle and the triangles (notice I said triangles--there's a hint).

3. ## Re: Diameter of circle--equilateral triangle

I should have said that this is not homework. I am 71 years old. If I had a compass I would try to figure it out geometrically as follows:

Bisect a side, set the compass to one half the side, draw an arc. Do the same thing with an adjacent--oops, that's not going to work. I would have to have the circle which is what I am trying to find.

Well, I've read some similar discussions (below) and apparently this can't be solved just using geometry--you have to use Pythagorean theorem to solve for r. Thanks anyway.

4. ## Re: Diameter of circle--equilateral triangle

If we deconstruct the equilateral triangle into 3 congruent isosceles 30°-30°-120° triangles, we find the two equal sides of these isosceles triangles is the radius of the circle.

Then, we may find the radius r in inches is:

$r=2\sec(30^{\circ})=\frac{4}{\sqrt{3}}$

And so the diameter d is:

$d=2r=\frac{8}{\sqrt{3}}\approx4.62\text{ in}$.

5. ## Re: Diameter of circle--equilateral triangle

Originally Posted by Jambe
I should have said that this is not homework. I am 71 years old. If I had a compass I would try to figure it out geometrically as follows:
Bisect a side, set the compass to one half the side, draw an arc. Do the same thing with an adjacent--oops, that's not going to work. I would have to have the circle which is what I am trying to find.

Well, I've read some similar discussions (below) and apparently this can't be solved just using geometry--you have to use Pythagorean theorem to solve for r. Thanks anyway.

You want the radius of the inscribed circle. It is right there.

6. ## Re: Diameter of circle--equilateral triangle

Originally Posted by Plato

You want the radius of the inscribed circle. It is right there.

(I wanted the circumscribed circle, not inscribed.)

The square root of 3, divided by 3, times 4"; times 2 for diameter.

Thanks everyone.

7. ## Re: Diameter of circle--equilateral triangle

Originally Posted by Jambe