What is the formula for calculating the distance between two points of an equilateral triangle with a known "radius"?

Question

I have an equilateral triangle with each point being a known distance of N units from the center of the triangle.

What formula would I need to use to determine the length of any side of the triangle?

Answer 1

Hint

let $L$ be the length of the triangle sides.

then

$$L^2=N^2+N^2-2N.N.\cos(\frac{2\pi}{3})$$

and

$$L=2N\sin(\frac{\pi}{3})=N\sqrt{3}$$

Answer 2

First draw the lines from all three points to the center point of this triangle, then notice that two if these lines make an isosceles triangle with angles $2\pi/3$, $\pi/6$, and $\pi/6$. Since we're given the length of two of the sides of this triangle, and the angles are known, we can calculate the length of the third side

Let $a$ denote the length of the inradius, and $b$ the sidelength if the triangle. Then, by the cosine law, we have

$b^2 = a^2 + a^2 - 2aa\cos(2\pi/3)$

$= 2a^2(1 - \cos(2\pi/3)$

$= 2a^2(3/2) $

So we have $b = \sqrt(3a^2)$ or $\sqrt3a$

Answer 3

If you label the triangle $ABC$ (from bottom left to right and top point $C$). Take the center to be $M$. Then consider the triangle formed by $AM$ and the middle of $AB:=Q$. So now you have $AMQ$ with angle $AMQ = 90$ degrees, and angle $MAQ = 60$ degrees. This means you have a $1:2:\sqrt{3}$ triangle with $|AM|=N$. This should be enough to solve it.