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?
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.
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:
$\displaystyle r=2\sec(30^{\circ})=\frac{4}{\sqrt{3}}$
And so the diameter d is:
$\displaystyle d=2r=\frac{8}{\sqrt{3}}\approx4.62\text{ in}$.
This page answers your question.
You want the radius of the inscribed circle. It is right there.