Why is the radius of an inscribed circle of a hexagon the height of the triangles that make the hexagon?

[Ana.stasia]

One question I had is how to calculate the volume of a pyramid which has a base that is a hexagon (all sides equal), an inscribed circle (for the base) and radius r. Hexagon is made out of six triangles and I got an idea to see what I would get if I treated r as the height of one of those triangles. I got the correct solution. However, I am not sure why r=h.

In case of a language barrier because I am still learning how to say certain math term in English, here is the picture I am describing.



Why can I be sure r is at 90 degrees on a and therefore divides the triangle shown into two same pieces? What is the rule here? Hope I explained it good.

Thank you in advance

 

[Dr.Peterson]

Any radius of a circle is perpendicular to the tangent at that point (which in this case is a side of the hexagon).

 

[pka]

One question I had is how to calculate the volume of a pyramid which has a base that is a hexagon (all sides equal), an inscribed circle (for the base) and radius r. Hexagon is made out of six triangles and I got an idea to see what I would get if I treated r as the height of one of those triangles. I got the correct solution. However, I am not sure why r=h.

In case of a language barrier because I am still learning how to say certain math term in English, here is the picture I am describing.

View attachment 22015

Why can I be sure r is at 90 degrees on a and therefore divides the triangle shown into two same pieces? What is the rule here? Hope I explained it good.

Because this is regular hexagon each angle of the sub-triangles measures \(\dfrac{\pi}{3}\) thus each such triangle is equilateral.