Question

Four congruent semicircles are drawn on the surface of a sphere with radius $2,$ as shown, creating a closed curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}.$ What is $n?$

$32$

$12$

$48$

$36$

$27$

Solution:

The curve is four congruent semicircular arcs, so its length is $4$ times one semicircle, $\pi r,$ where $r$ is the arc radius. The arcs meet at four points that form a square inscribed in a great circle of the radius- $2$ sphere, and each arc's diameter is a side of that square, a chord of length $2\sqrt2.$ So $r = \sqrt2.$ (Check it another way: the small circle sits in a plane at distance $\frac{2}{\sqrt2} = \sqrt2$ from the center, giving radius $\sqrt{2^2 - (\sqrt2)^2} = \sqrt2.$ ) The total length is $4 \cdot \pi\sqrt2 = \pi\sqrt{32},$ so $n = 32.$ Therefore, the answer is A.

2023 AMC 10B Problem 20

Solution:

Problem 20 in Other Years