Question

The following figure is made of six semicircles with the same diameter, and of four congruent equilateral triangles represented by AEG, EGF, BEF, and GFC. If AE = 3cm, then the area of this figure is, in cm2, equal to:

A)
`9(3\pi + 4√(3) )`
B)
`4(3\pi + 4√(3) )`
C)
`(9)/(4)(3\pi + 4√(3))`
D)
`(4)/(9)(3\pi + 4√(3))`
Answer: "C."
Why is it letter "C"?
No matter what I try, I cannot obtain this result. Does anyone know the correct procedure for this?

Answer 1

6 semicircles = 3 circles

The formula of the area of a equilateral triangle:

`A_\triangle=(a^2\sqrt3)/(4)`

We have a = |AE| = 3cm. Substitute:

`A_\triangle=(3^2\sqrt3)/(4)=(9\sqrt3)/(4)\ cm^2`

The formula of the area of a circle:

`A_O=\pi r^2`

We have 2r = |AE| =3cm → r = 1.5cm. Substitute:

`A_O=\pi\cdot1.5^2=2.25\pi\ cm^2`

The area of the figure:

`A=4A_\triangle+3A_O\\\\A=4\cdot(9\sqrt3)/(4)+3\cdot2.25\pi=9\sqrt3+6.75\pi=6(3)/(4)\pi+9\sqrt3\\\\=(27)/(4)\pi+(36\sqrt3)/(4)=(9)/(4)\cdot3\pi+(9)/(4)\cdot4\sqrt3=(9)/(4)(3\pi+4\sqrt3)\ cm^2`