How to calculate the Volume of a Frustum
What is a Frustum
A frustum is a geometric shape formed by cutting the top portion of a cone or pyramid with a plane parallel to the base.
Understanding how to calculate the volume of a frustum is useful in various real-world applications, such as architecture, engineering, and manufacturing.
How to calculate the Volume of a Frustum Step-by-Step
To calculate the volume of a frustum, follow these steps:
Identify the measurements required. You need the dimensions of the two bases (smaller and larger) and the height of the frustum.
Find the height of the complete cone or pyramid (before the top was cut off) and the height of the small piece that was removed. The two solids are similar, so the ratio of the base dimensions equals the ratio of the heights. With small radius r, large radius R and frustum height h, the small cone height satisfies r/R = (small height) / (small height + h); the large cone height is the small height plus h. Do not use the frustum height as the height of either cone.
Calculate the volume of the large (complete) cone or pyramid. Use (1/3) * base area * its full height, with the larger base and the full height found in step 2.
Calculate the volume of the small (removed) cone or pyramid. Use (1/3) * base area * its height, with the smaller base and the smaller height found in step 2.
Subtract the volume of the small cone or pyramid from the volume of the large cone or pyramid. The result is the volume of the frustum.
Shortcut check: a frustum of height h whose two parallel faces have areas A and B (with B the larger) has volume (h/3) * (A + B + the square root of A * B). This gives the same answer without finding the cone heights, so it is a useful way to check your working.
Worked Examples:
Let’s work through a couple of examples to illustrate the process of calculating the volume of a frustum:
Example 1: A frustum has a smaller circular base with a radius of 4 cm, a larger circular base with a radius of 8 cm, and a height of 10 cm. Calculate the volume of the frustum.
Solution:
- Areas of the bases: smaller = π * (4 cm)^2 = 16π cm^2; larger = π * (8 cm)^2 = 64π cm^2.
- Find the cone heights (similar triangles). With small height H, the cones are similar so 4/8 = H / (H + 10), giving H = 10 cm. So the small cone is 10 cm tall and the complete (large) cone is 10 + 10 = 20 cm tall.
- Volume of the large cone = (1/3) * π * (8 cm)^2 * 20 cm = 1280π/3 cm^3 ≈ 426.667π cm^3
- Volume of the small cone = (1/3) * π * (4 cm)^2 * 10 cm = 160π/3 cm^3 ≈ 53.333π cm^3
- Volume of the frustum = large cone – small cone = 1280π/3 – 160π/3 = 1120π/3 cm^3 ≈ 373.333π cm^3
So the volume of the frustum is 1120π/3 cm^3, which is about 373.333π cm^3 ≈ 1173 cm^3.
Check with the frustum formula (h/3) * (A + B + the square root of A * B) = (10/3) * (16π + 64π + the square root of 16π * 64π) = (10/3) * (80π + 32π) = (10/3) * 112π = 1120π/3 cm^3. This matches.
Example 2: A frustum has a smaller square base with a side length of 6 cm, a larger square base with a side length of 12 cm, and a height of 15 cm. Calculate the volume of the frustum.
Solution:
- Areas of the bases: smaller = (6 cm)^2 = 36 cm^2; larger = (12 cm)^2 = 144 cm^2.
- Find the pyramid heights (similar triangles). With small height H, 6/12 = H / (H + 15), giving H = 15 cm. So the small pyramid is 15 cm tall and the complete (large) pyramid is 15 + 15 = 30 cm tall.
- Volume of the large pyramid = (1/3) * 144 cm^2 * 30 cm = 1440 cm^3
- Volume of the small pyramid = (1/3) * 36 cm^2 * 15 cm = 180 cm^3
- Volume of the frustum = large pyramid – small pyramid = 1440 – 180 = 1260 cm^3
So the volume of the frustum is 1260 cm^3.
Check with the frustum formula (h/3) * (A + B + the square root of A * B) = (15/3) * (36 + 144 + the square root of 36 * 144) = 5 * (180 + 72) = 5 * 252 = 1260 cm^3. This matches.