Volume of a Frustum
Calculate the volume of a frustum (truncated cone or pyramid).
Learn the formula with step-by-step examples for both shapes.
The Formulas
Pyramid frustum: V = (h/3)(A₁ + A₂ + √(A₁ × A₂))
A frustum is the portion of a solid (usually a cone or pyramid) that lies between two parallel planes cutting through it. Imagine taking a cone and slicing off the top with a horizontal cut — the remaining bottom portion is a frustum.
Frustums are extremely common in the real world. Buckets, lampshades, drinking cups, volcanic craters, and building foundations are all frustum-shaped. Even the ancient Egyptians used the frustum volume formula, as documented in the Moscow Mathematical Papyrus from around 1850 BCE.
For a cone frustum, R is the radius of the larger base and r is the radius of the smaller base. If r = 0, the formula simplifies to the volume of a complete cone: V = πhR²/3. If r = R, the frustum becomes a cylinder: V = πhR². This shows that the frustum formula gracefully handles both limiting cases.
Variables
| Symbol | Meaning |
|---|---|
| V | Volume of the frustum |
| h | Height (perpendicular distance between the two bases) |
| R | Radius of the larger base (for cone frustum) |
| r | Radius of the smaller base (for cone frustum) |
| A₁ | Area of the larger base (for pyramid frustum) |
| A₂ | Area of the smaller base (for pyramid frustum) |
Example 1
A bucket has a bottom radius of 10 cm, a top radius of 15 cm, and a height of 25 cm. What is its volume?
Identify values: R = 15 cm, r = 10 cm, h = 25 cm
V = (πh/3)(R² + Rr + r²) = (π × 25 / 3)(225 + 150 + 100)
V = (78.54 / 3)(475) = 26.18 × 475
V ≈ 12,435 cm³ (about 12.4 liters)
Example 2
A pyramid-shaped building foundation has a square base of 20 m × 20 m at the bottom and 16 m × 16 m at the top, with a height of 3 m. What is the concrete volume?
A₁ = 20 × 20 = 400 m², A₂ = 16 × 16 = 256 m², h = 3 m
V = (h/3)(A₁ + A₂ + √(A₁ × A₂))
√(400 × 256) = √102,400 = 320
V = (3/3)(400 + 256 + 320) = 1 × 976
V = 976 m³ of concrete
When to Use It
The frustum formula is used in many practical applications.
- Calculating the capacity of tapered containers like buckets and cups
- Estimating concrete volume for foundations and retaining walls
- Engineering calculations for hoppers and funnels
- Calculating earthwork volumes for road and dam construction