What is the area of a hexagon inscribed in a circle if the diameter is 23 inches?

Question

note: the hexagon can be subdivided into 6 congruent equilateral triangles

Answer 1

Since the dia=23"; the rad=11.5; this is side "c" of the triangles the hexagon is made of.

The 6 triangles are to be cut in half; they'll become 12 rt triangles.

The center becomes 12 30° angles (12*30=360°). These angles are labeled "α" (alpha).

The sides that make a rt angle with the side that has both ends on the circle are sides "b".

" " runs from the center to the angles at the circle are sides "c" (hypotenuse).

Sides "a" connect sides "b" and "c".

A formula used to get lengths is a^2+b^2=c^2.

In this case we still need one more side.

Another formula is a/sinα=b/sinβ(beta)=c/sinγ(gamma).

We have α=30°. γ is the rt angle; so β=60°.

Since the rad=11.5"= c/sinγ because the sin of a rt angle=1.

So a/sinα=c; a/sin30°=a/.5; a=5.75".

Another way to set the formula is c^2-b^2=a^2.

So 11.5^2-5.75^2=99.188^.5=9.96'.

Area=a*b*6; 5.75*9.96*6=343.6sq/in.

Answer 2

Area = 6*(1/2)*(11.5)²*sin 60°