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) $9(3\pi + 4\sqrt{3} )$
B) $4(3\pi + 4\sqrt{3} )$
C) $\frac{9}{4}(3\pi + 4\sqrt{3})$
D) $\frac{4}{9}(3\pi + 4\sqrt{3})$
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:

$A_\triangle=\dfrac{a^2\sqrt3}{4}$

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

$A_\triangle=\dfrac{3^2\sqrt3}{4}=\dfrac{9\sqrt3}{4}\ cm^2$

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:

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