find the area of the hexagon with a radius of 10

Question

A.75√3

B.300

C. 150√3

Answer 1

Assuming a regular hexagon inscribed inside of a circle !

There are 6 congruent triangles. Take one of them

Draw the altitude and call it h.

Call the radius r.

Call half of a side x.

Call the top right-side angle θ.

cosθ=h/r and sinθ=x/r

h=rcosθ and x=rsinθ

A=(1/2)(b)(h)

(1/2)(b)=x

A=xh

substituting

A=rsinθrcosθ

A=r²sinθcosθ

from trig identity sin2θ=2sinθcosθ

A=r²(1/2)sin2θ

The angle 2θ at the apex is determined by the number of sides, n

Therefore 360/n=2θ

A=(1/2)(r²)sin(360/n)

A=(1/2)(10²)sin(360/6)

A=(1/2)(100)(sin60º)

A=50(√3/2)

A=25√3 for one of six congruent triangles

multiply this area by 6 to get the area of the hexagon

A=6*25√3

A=150√3 square units which is choice C.