Conical frustum calculator
Volume, slant height and surfaces of a truncated cone from two radii and the height.
Fill in the fields and the result will appear here automatically.
A right circular frustum has concentric parallel bases and a perpendicular height. Its volume V = πh(R²+Rr+r²)/3 accounts for a radius changing continuously with height. Slant height l = √(h²+(R−r)²) follows the lateral surface; substituting h for l understates that area. This tool uses R > r ≥ 0: enter the larger radius first, while a zero smaller radius gives a full cone.
How it works
Formula and logic
Volume V = πh(R² + Rr + r²)/3. Slant height l = √(h² + (R − r)²). The lateral surface is π(R + r)l, and the total surface adds both bases.
Example
A frustum with radii of 6 and 3 cm and a height of 8 cm has a volume of 527.79 cm³.
Fields and units
- Length unit — list option
- Larger radius — cm
- Smaller radius — cm
- Height — cm
How to use
- — Choose the length unit.
- — Enter the larger base radius R.
- — Enter the smaller radius r regardless of the solid’s orientation.
- — Enter the height: the vertical distance between the bases, not the slant.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: volume by integrating cross-sectional areas OpenStax: sine and tangent in a right triangle
- Limitation
- A right circular frustum with parallel circular bases and one common axis. Total surface includes both bases; wall thickness, rim and partial filling are not modelled. Capacity needs internal dimensions in one selected unit.
FAQ
How is the height different from the slant height?
Height h is perpendicular to the bases; slant l follows the surface. With unequal radii l > h, so π(R+r)h is smaller than the correct lateral area π(R+r)l.
Where does the Rr term in the volume come from?
Cross-section radius changes linearly, while its area is quadratic. Integrating π[R+(r−R)z/h]² from 0 to h gives πh(R²+Rr+r²)/3. Averaging the base areas instead overstates volume by πh(R−r)²/6.
What happens if the top radius is zero?
You get an ordinary cone, and the formula reduces to πR²h/3. That makes a convenient check on the result.
Why must the top radius be smaller than the bottom one?
That is this tool’s input order, not a restriction on how a solid can be oriented. Enter the larger radius as R and the smaller as r regardless of how the vessel stands. Equal radii are excluded here; use the cylinder calculator for that case.
How do I work out what a bucket holds?
Enter the radii of the base and the rim and the inside height. The volume in cubic centimetres divided by 1000 gives litres.