Area & Volume - SAT Math

$A = lw$.

$A = s^2$.

$A = \tfrac{1}{2}(10)(6) = 30$.

$A = \tfrac{1}{2}d_1 d_2$.

$A = \tfrac{1}{2}(b_1 + b_2)h$.

$P = 2(l + w)$.

$P = 4s$.

$A = \pi r(r + l)$, where $l$ is slant height.

$A = 6s^2$.

$A = 2\pi r^2 + 2\pi rh$.

$A = 2(lw + lh + wh)$.

$A = 4\pi r^2$.

$V = \tfrac{1}{3}\pi r^2 h$.

$V = s^3$.

$V = \pi r^2 h$.

$V = \tfrac{1}{3}Bh$, where $B$ is base area.

$V = lwh$.

$V = \tfrac{4}{3}\pi r^3$.

Surface area increases by a factor of 4.

Volume increases by a factor of 8.

Let's call the radius of the sphere r. The formula for the surface area of a sphere (A) is given below:

A = 4_πr_2

Because the sphere is inscribed inside the cube, the diameter of the sphere is equal to the side length of the cube. Because the diameter is twice the length of the radius, the diameter of the sphere is 2_r_. This means that the side length of the cube is also 2_r_.

The surface area for a cube is given by the following formula, where s represents the length of each side of the cube:

surface area of cube = 6_s_2

The formula for surface area of a cube comes from the fact that each face of the cube has an area of _s_2, and there are 6 faces total on a cube.

Since we already determined that the side length of the cube is the same as 2_r_, we can replace s with 2_r_.

surface area of cube = 6(2_r_)2 = 6(2_r_)(2_r_) = 24_r_2.

We are asked to find the ratio of the surface area of the cube to the surface area of the sphere. This means we must divide the surface area of the cube by the surface area of the sphere.

ratio = (24_r_2)/(4_πr_2)

The _r_2 term cancels in the numerator and denominator. Also, 24/4 simplifes to 6.

ratio = (24_r_2)/(4_πr_2) = 6/π

The answer is 6/π.