Question

Suppose the height of a cylinder is equal to its radius. The cylinder can fit inside a square prism, as shown below.

The cross-sectional areas are still the same and the ratio of the area of the circle to the area of the square is still StartFraction pi Over 4 EndFraction.

StartFraction pi Over 4 EndFraction
Complete the derivation of the formula for a cylinder whose height is equal to its radius.



The prism’s volume is the area of the base,

, times the height,
.



Since the ratio of the areas is StartFraction pi Over 4 EndFraction, then the volume of the cylinder is
times the volume of the prism.



V = A cylinder inside of a square prism is shown. The cylinder has a height and radius with a length of r. The length of the square prism is 2 r.(4r3), or

Answer 1

Let the cylinder have radius r and height h = r. The square prism that just contains it has side length 2r.

1. Area of square base = (2r)^2 = 4r^2.
2. Prism volume = base area × height = 4r^2 · r = 4r^3.
3. The circle’s area is (π/4) of the square’s area, so the cylinder’s volume is (π/4) of the prism’s volume:
V = (π/4) · 4r^3 = πr^3.

This agrees with V = πr^2h since h = r: V = πr^2·r = πr^3.