[Math] How to calculate the radius of a circle inscribed in a regular hexagon

circlesgeometry

If I know how long one side of a regular hexagon is, what's the formula to calculate the radius of a circle inscribed inside it?

Illustration:

in-circle in a hexagon

Best Answer

Label the center of the circle. Draw six lines from the the center to the circle to the vertices of the hexagon. (These lines will be longer than the radius.) This will divide the circle into six triangles.

Question for you: Tell me every thing you can about these triangles. In particular, what are the lengths of the lines from the center?

Now draw six radii of the circle to the six edges of the hexagon. Along with the six "spokes" before you have divided the hexagon into twelve triangles.

Question for you: tell me every thing you can about these triangles. In particular:

are they congruent to each other?

what are the angles of these triangles?

What are the lengths of the sides of these triangles?

And from there I will ask you these two questions: What is the radius of the circle? and, what is the formula for the area of the circle.

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