Volume of a frustum (truncated cone and pyramid)

A frustum is what you get when you cut the top off a cone or a pyramid. Instead of a point there is now a small end and a big end. Both kinds have a volume formula, and both formulas can look a bit difficult, so let's take them one at a time.

When do I use this?

Rarely, honestly. Many of the formulas in this part of geometry you will probably never use, and the frustums are the ones you meet least. But when a task does give you a bucket shape or a cut-off pyramid, this is the formula. The general idea of volume is explained in 3D shapes: volume and surface area.

The frustum of a cone

The volume of a frustum of a cone can be found if we have the height and the radius of the small circle and of the big circle. We call the big radius $r_1$ and the small radius $r_2$, and $h$ is the height straight up between the two circles. We use the formula:

$$V = \frac{1}{3} \times \pi \times h \times (r_2^2 + r_1^2 + r_2 \times r_1)$$

Take a frustum with height $6$ cm, a big circle of radius $4$ cm and a small circle of radius $2$ cm. Inside the bracket:

$$r_2^2 + r_1^2 + r_2 \times r_1 = 2^2 + 4^2 + 2 \times 4 = 4 + 16 + 8 = 28$$

And then

$$V = \frac{1}{3} \times \pi \times 6 \times 28 = 56\pi \approx 175.93 \text{ cm}^3$$

So about $175.93$ cubic centimetres fit inside it.

The frustum of a pyramid

For a pyramid with the top cut off we need the height and the areas of the two ends. We call the area of the big end $g_1$ and the area of the small end $g_2$. Then we can find the volume with the formula

$$V = \frac{1}{3} \times h \times (g_2 + g_1 + \sqrt{g_2 \times g_1})$$

Take a frustum of a square pyramid with height $4$ cm, a big end of $36$ cm² and a small end of $9$ cm². Inside the bracket:

$$g_2 + g_1 + \sqrt{g_2 \times g_1} = 9 + 36 + \sqrt{9 \times 36} = 9 + 36 + \sqrt{324} = 9 + 36 + 18 = 63$$

And then

$$V = \frac{1}{3} \times 4 \times 63 = 84 \text{ cm}^3$$

So $84$ cubic centimetres fit inside it.

What about the surface area?

Finding the surface area of a frustum can be quite involved. For the pyramid frustum we have to find the area of the two ends, and then add the area of the sides. How the pyramid looks has a big effect on how we work that out, so you end up having to split the shape into smaller pieces.

In most tasks you will not meet the surface area of a frustum, because they are quite hard. So we leave it there.

Common mistakes

• Mixing up which end is which. The formulas need both ends, and each radius or area goes in its own place. It helps to write down $r_1$ and $r_2$, or $g_1$ and $g_2$, before you start.
• Skipping the middle term in the bracket. The bracket has three parts: the small end, the big end, and the mixed term ($r_2 \times r_1$ or $\sqrt{g_2 \times g_1}$). All three are needed.
• Giving the answer in the wrong unit. A volume is in cubic units, cm³. If it comes out in cm², we have probably calculated wrong somewhere.

Related

The cone and pyramid these come from are in volume and surface area of a cone and volume of a pyramid. The overview of every shape, and the reassurance about which formulas you actually need, is in 3D shapes: volume and surface area.