Volume of Cylinders, Cones, and Spheres

In geometry, finding the volume of three-dimensional shapes tells us how much space is inside them. For shapes with curved surfaces—like cylinders, cones, and spheres—the formulas all involve $\pi$ (pi) and the radius ($r$) of the shape.

Volume of a Cylinder

A cylinder is essentially a circular prism. To find its volume, you calculate the area of its circular base ($\pi r^2$) and multiply it by its height ($h$).

Formula: $V = \pi r^2 h$

Example: Find the volume of a cylinder with radius $3$ and height $10$.

1. Identify the given values: $r = 3$, $h = 10$.
2. Plug them into the formula: $V = \pi (3)^2 (10)$.
3. Calculate the exponent: $3^2 = 9$.
4. Multiply: $V = \pi (9)(10) = 90\pi$.

The volume is $90\pi$ cubic units.

Volume of a Cone

A cone is closely related to a cylinder. If a cone and a cylinder have the exact same base radius and height, the cone will hold exactly one-third of the volume of the cylinder.

Formula: $V = \frac{1}{3}\pi r^2 h$

Example: Find the volume of a cone with radius $4$ and height $9$.

1. Identify the given values: $r = 4$, $h = 9$.
2. Plug them into the formula: $V = \frac{1}{3}\pi (4)^2 (9)$.
3. Calculate the exponent: $4^2 = 16$.
4. Multiply: $V = \frac{1}{3}\pi (16)(9) = \pi (16)(3) = 48\pi$.

The volume is $48\pi$ cubic units.

Volume of a Sphere

A sphere is a perfectly round 3D object, like a basketball. Unlike cylinders and cones, a sphere doesn't have a height. Its volume depends entirely on its radius.

Formula: $V = \frac{4}{3}\pi r^3$

Example: Find the volume of a sphere with radius $5$.

1. Identify the radius: $r = 5$.
2. Plug it into the formula: $V = \frac{4}{3}\pi (5)^3$.
3. Calculate the cube: $5^3 = 5 \times 5 \times 5 = 125$.
4. Multiply: $V = \frac{4}{3}\pi (125) = \frac{500}{3}\pi$.

The volume is $\frac{500}{3}\pi$ cubic units.