Volume

Volume: Measuring Interior Space

Volume measures how much three-dimensional space a solid takes up, in cubic units. For prisms and cylinders, there's a unifying idea: volume equals the area of the base multiplied by the height, since you're essentially stacking identical copies of the base shape upward.

Rectangular Prisms

For a rectangular prism with length l, width w, and height h, the base is a rectangle of area l\times w, so:

For a cube with edge length s, this simplifies to V = s^3.

Cylinders

A cylinder's base is a circle of area \pi r^2, so stacking that circular base up to height h gives:

Cones and Spheres

Picture a cone as an ice cream cone: a circular base of radius r tapering to a single point (the apex) at height h. A cone's volume is exactly one-third of a cylinder with the same base and height:

A sphere (a perfectly round ball) of radius r has volume:

These last two formulas are worth memorizing directly, since deriving them requires calculus — just remember the \frac{1}{3} for cones and the \frac{4}{3} for spheres.

Example

A cylindrical water tank has radius 4 and height 9. Find its volume, and then find the volume of a cone with the same radius and height for comparison.

1. Cylinder volume: V = \pi r^2 h = \pi (4)^2(9) = \pi(16)(9) = 144\pi
2. Cone volume with the same base and height: V = \frac{1}{3}\pi r^2 h = \frac{1}{3}(144\pi) = 48\pi
3. As expected, the cone's volume (48\pi) is exactly one-third of the cylinder's volume (144\pi).

Key Takeaways

• Rectangular prism: V = lwh; cube: V=s^3.
• Cylinder: V = \pi r^2 h (circular base area times height).
• Cone: V = \frac{1}{3}\pi r^2 h — one-third of the matching cylinder.
• Sphere: V = \frac{4}{3}\pi r^3.