Problem: Two equal parallel chords are drawn 8 88 inches apart in a circle of radius 8 88 inches. The area of that part of the circle that lies between the chords is:
Answer Choices:
A. 21 1 3 π − 32 3 21 \dfrac{1}{3} \pi-32 \sqrt{3}2 1 3 1 ​ π − 3 2 3 ​
B. 32 3 + 21 1 3 π 32 \sqrt{3}+21 \dfrac{1}{3} \pi3 2 3 ​ + 2 1 3 1 ​ π
C. 32 3 + 42 2 3 π 32 \sqrt{3}+42 \dfrac{2}{3} \pi3 2 3 ​ + 4 2 3 2 ​ π
D. 16 3 + 42 2 3 π 16 \sqrt{3}+42 \dfrac{2}{3} \pi1 6 3 ​ + 4 2 3 2 ​ π
E. 42 2 3 π 42 \dfrac{2}{3} \pi4 2 3 2 ​ π
Solution:
By symmetry, the required area is 4 ( T + S ) 4(T+S)4 ( T + S ) .
T = 1 2 4 ⋅ 4 3 = 8 3 , S = ( 30 / 360 ) ⋅ π 8 2 = 5 1 3 π ∴ A = 32 3 + 21 1 3 π . \begin{aligned}
& T=\dfrac{1}{2} 4 \cdot 4 \sqrt{3}=8 \sqrt{3}, \\
& S=(30 / 360) \cdot \pi 8^{2}=5 \dfrac{1}{3} \pi \\
& \therefore A=32 \sqrt{3}+21 \dfrac{1}{3} \pi .
\end{aligned}
​ T = 2 1 ​ 4 ⋅ 4 3 ​ = 8 3 ​ , S = ( 3 0 / 3 6 0 ) ⋅ π 8 2 = 5 3 1 ​ π ∴ A = 3 2 3 ​ + 2 1 3 1 ​ π . ​