Volume Formulas for Common 3D Shapes

Here are the main volume formulas for common 3D shapes: prisms and cylinders use base area times height, pyramids and cones use one-third of that pattern, and spheres use a radius-based formula. Once you see that structure, the formulas are easier to understand and remember.

Volume Formulas for Common 3D Shapes

Solid	Volume formula	What to know
Rectangular prism	$V = lwh$	Length, width, and height
Cube	$V = s^3$	All edges have the same length
Any prism	$V = Bh$	$B$ is the area of the base
Cylinder	$V = \pi r^2 h$	Same as $Bh$ because the base is a circle
Any pyramid	$V = \frac\{1\}\{3\}Bh$	One-third of a prism with the same base and height
Cone	$V = \frac\{1\}\{3\}\pi r^2 h$	One-third of a cylinder with the same base and height
Sphere	$V = \frac\{4\}\{3\}\pi r^3$	Uses radius, not height

For pyramids and cones, $h$ means the perpendicular height. If a problem gives a slant height instead, that number does not go directly into the volume formula.

Why Most Volume Formulas Follow the Same Pattern

The simplest idea is this:

$$V = Bh$$

Here, $B$ means the area of the base, and $h$ is the height measured straight up from that base.

That one pattern explains several formulas at once. A rectangular prism uses a rectangular base, so $B = lw$ and the formula becomes $V = lwh$. A cylinder uses a circular base, so $B = \pi r^2$ and the formula becomes $V = \pi r^2 h$.

Pyramids and cones use the same base and height idea, but only one-third as much volume as the matching prism or cylinder:

$$V = \frac{1}{3}Bh$$

The sphere is the main common solid that does not fit the base-times-height pattern, which is why its formula is worth remembering separately.

Worked Example: Finding the Volume of a Cone

Find the volume of a cone with radius $3$ cm and height $8$ cm.

Use the cone formula:

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

Substitute the values:

$$V = \frac{1}{3}\pi (3^2)(8)$$

Simplify:

$$V = \frac{1}{3}\pi (9)(8) = \frac{72}{3}\pi = 24\pi$$

So the volume is $24\pi\ \text{cm}^3$, which is about $75.4\ \text{cm}^3$.

This example is useful because the matching cylinder with the same radius and height would have volume $72\pi\ \text{cm}^3$. The cone is exactly one-third of that, which is a good built-in check.

Common Mistakes With Volume Formulas

1. Using diameter where the formula expects radius. If you are given $d$, convert first with $r = \frac{d}{2}$.
2. Using slant height for a cone or pyramid. Volume uses perpendicular height.
3. Mixing up surface area and volume. Volume answers how much space is inside, not how much outside covering there is.
4. Forgetting cubic units. Volume should be written as units like $\text{cm}^3$, $\text{m}^3$, or $\text{in}^3$.
5. Treating $B$ as a side length instead of base area. In $V = Bh$, $B$ is already an area.

When to Use Volume Formulas

Volume formulas are used when you need the capacity or internal size of a 3D object. In class, that usually means geometry problems. Outside class, the same idea appears when estimating how much a box can hold, how much liquid fits in a tank, or how much material fills a container.

The condition matters: the formula is only as accurate as the shape model. If a real object is only approximately cylindrical or spherical, the result is also an approximation.

Try Your Own Version

Pick a cylinder with radius $4$ units and height $10$ units, then find the volume. After that, keep the same base and height but switch to a cone. Seeing those two answers side by side is one of the fastest ways to make the formulas stick.