Frustum Volume Calculator

On This Page

• Conical frustum volume formula
• How to calculate volume of a frustum
• Frustum of a cone and special cases
• Pyramid, square, and rectangular frustums
• Worked examples
• Frustum volume units, liters, and gallons
• Frequently asked questions

Truncated Cone Volume Formula (Conical Frustum)

A frustum (truncated cone) is the part of a cone remaining after the top is cut off parallel to the base, leaving two circular faces of different sizes.

• R = larger radius; r = smaller radius, the formula is symmetric, so which you label top or bottom does not matter
• h = vertical height between the two circular faces (not slant height)
• If a diameter is given, divide by 2 first: r = d ÷ 2

The mixed term Rr is why averaging the two radii and using the cylinder formula gives the wrong answer, a frustum holds more volume near the larger end than a simple average captures. If one radius is zero, the shape is a standard cone; use the Cone Volume Calculator in that case.

How to Calculate Volume of a Frustum

Identify the two radii and the vertical height, then work through the formula in order. All three measurements must use the same unit.

1. Find R, r, and h (convert diameters to radii if needed)
2. Calculate R², Rr, and r²
3. Add the three terms: R² + Rr + r²
4. Multiply by πh ÷ 3
5. Write the answer in cubic units

Frustum of a Cone and Special Cases

A conical frustum is also called a frustum of a cone or truncated cone. It is what remains when the top of a cone is cut off parallel to the base.

• If r = 0: the frustum becomes a cone.
• If R = r: the frustum becomes a cylinder.
• If the top and bottom are circles: use the conical frustum formula on this page.

Frustum shapes show up often in hoppers, grain silos, and buckets. For a straight-sided tank with no taper, the Tank Volume Calculator gives the equivalent rectangular-tank formula.

Pyramid, Square, and Rectangular Frustums

A pyramid frustum volume calculator uses areas, not radii. This applies to frustums with square, rectangular, triangular, or polygonal parallel faces.

• A1 = area of one parallel face
• A2 = area of the other parallel face
• h = vertical height between the faces

For a square frustum, calculate each square area first. For a rectangular frustum, calculate the top rectangle area and bottom rectangle area first. A full (untruncated) pyramid is the special case where the top face shrinks to a point; the Pyramid Volume Calculator covers that case directly.

Worked Examples

Tapered shapes should not be treated like plain cylinders. Check whether the shape is circular or pyramid-based before choosing a formula.

Example 1: Conical Frustum Plant Pot

Problem: A plant pot is shaped like a frustum of a cone. The top radius is 9 cm, the bottom radius is 5 cm, and the inside height is 14 cm. Estimate the internal volume.

1. R = 9 cm, r = 5 cm, h = 14 cm
2. R² = 81, Rr = 45, r² = 25
3. Sum = 81 + 45 + 25 = 151
4. V = π × 14 × 151 ÷ 3
5. V ≈ 2,213.98 cm³

Answer: The plant pot volume is about 2,214 cm³.

Example 2: Frustum of a Cone From a Cut Cone

Problem: A cone is cut parallel to its base, leaving a frustum with bottom radius 12 cm, top radius 4 cm, and height 18 cm. Find the volume.

1. R² = 12² = 144
2. Rr = 12 × 4 = 48
3. r² = 4² = 16
4. V = π × 18 × (144 + 48 + 16) ÷ 3
5. V ≈ 3,920.71 cm³

Answer: The cone frustum volume is about 3,920.7 cm³.

Example 3: Diameter Given Instead of Radius

Problem: A tapered container has top diameter 20 inches, bottom diameter 12 inches, and vertical height 15 inches. Find the volume.

1. Top radius = 20 ÷ 2 = 10 in
2. Bottom radius = 12 ÷ 2 = 6 in
3. R² + Rr + r² = 10² + 10 × 6 + 6² = 196
4. V = π × 15 × 196 ÷ 3
5. V ≈ 3,078.76 in³

Answer: The frustum volume is about 3,078.8 cubic inches.

Example 4: Square Pyramid Frustum

Problem: A square frustum has bottom side 10 cm, top side 6 cm, and vertical height 8 cm. Find the volume.

1. Bottom area A1 = 10² = 100 cm²
2. Top area A2 = 6² = 36 cm²
3. √(A1A2) = √(100 × 36) = 60
4. V = 8(100 + 36 + 60) ÷ 3
5. V = 522.67 cm³

Answer: The square frustum volume is about 522.7 cm³.

Example 5: Frustum Capacity in Liters

Problem: A conical frustum tank has bottom radius 0.8 m, top radius 0.5 m, and height 1.2 m. Estimate its capacity in liters.

1. R² + Rr + r² = 0.8² + 0.8 × 0.5 + 0.5² = 1.29
2. Volume = π × 1.2 × 1.29 ÷ 3
3. Volume ≈ 1.621 m³
4. Liters = 1.621 × 1,000 = 1,621 L

Answer: The frustum tank holds about 1,621 liters.

Frustum Volume Units, Liters, and Gallons

Frustum volume is written in cubic units. If the dimensions are in centimeters, the result is cm³. If the dimensions are in meters, the result is m³.

Frequently Asked Questions

How do you calculate the volume of a frustum?

For a conical frustum: V = (πh / 3)(R² + Rr + r²), where R is the larger radius, r is the smaller radius, and h is vertical height. For a pyramid frustum: V = h(A1 + A2 + √(A1 × A2)) / 3, where A1 and A2 are the two parallel face areas.

Is a frustum the same as a truncated cone?

Yes, same shape, different names. "Truncated cone" is common in engineering and manufacturing; "conical frustum" is the formal geometric term. Both refer to a cone with the top cut off parallel to the base.

Is my bucket really a frustum?

Almost certainly. Standard plastic and metal buckets taper wider toward the top so they can stack. That tapered shape is exactly a conical frustum. Measure the top radius, bottom radius, and inside height and run them through this calculator to find the real liquid capacity rather than relying on the label.

Why can't I just average the two radii and use the cylinder formula?

Averaging underestimates volume. The frustum formula uses R² + Rr + r² rather than ((R+r)/2)² because the taper creates more volume near the larger end. The difference is small for gradual tapers but grows significant when R and r differ substantially.

Can top radius and bottom radius be switched?

Yes. The formula is symmetric in R and r, swapping labels does not change the result.

References

• Conical Frustum, Wolfram MathWorld: formal mathematical properties: volume formula, lateral surface area, and why the Rr term is necessary.
• Pyramidal Frustum, Wolfram MathWorld: the area-based volume formula for square, rectangular, and polygonal frustums.
• Frustum, Math is Fun: practical explanation with diagrams showing the difference between a frustum and a full cone.