Ratio between surface of a cube and sphere

Surface of a cube with an edge a a equals to the surface of a sphere with a radius r r . What must be the ratio a r \frac{a}{r} equal to?

((3 * pi)^(1/2)) / 4 ((3 * pi)^(1/2)) / 2 ((3 * pi)^(1/2)) / 6 ((6 * pi)^(1/2)) / 6 ((6 * pi)^(1/2)) / 3 ((8 * pi)^(1/2)) / 3

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2 solutions

The surface area of a cube is given by S c = 6 a 2 S_c=6a^2 where a a is the side length. The surface area of a sphere is given by S s = 4 π r 2 S_s=4\pi r^2 where r r is the radius. Since the surface areas are equal, we have

6 a 2 = 4 π r 2 6a^2=4\pi r^2

Dividing both sides by 6 r 2 6r^2 , we get

a 2 r 2 = 4 π 6 \dfrac{a^2}{r^2}=\dfrac{4\pi}{6}

Extracting the square root of both sides, we get

a r = 2 π 6 \dfrac{a}{r}=\dfrac{2\sqrt{\pi}}{\sqrt{6}}

Rationalizing the denominator by multiplying the above result by 6 6 \dfrac{\sqrt{6}}{\sqrt{6}} , we get

a r = 6 π 3 \dfrac{a}{r}=\boxed{\dfrac{\sqrt{6\pi}}{3}}

Tomáš Hauser
Jun 17, 2018

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