Frustum Volume Calculator
Calculate the volume of a frustum (truncated cone) using top radius, bottom radius, and height. Formula: V = (pi x h / 3) x (R^2 + R*r + r^2).
How to use this tool
- Enter top radius (r), bottom radius (r) and height in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your volume and the full breakdown beneath it.
Formula
V = (pi x h / 3) x (R^2 + R*r + r^2)
How it works
A frustum is a cone with its tip cut off: V = ⅓πh(R² + Rr + r²), where R is the larger base radius and r is the smaller top radius.
Worked example
Frustum R=6, r=3, h=10
- R=6, r=3, h=10
- R^2 + R*r + r^2 = 36 + 18 + 9 = 63
- V = (pi x 10 / 3) x 63 = 210 x pi
- V = 659.7344...
Volume = 659.7345 cubic units
Common mistakes to avoid
- Substituting 0 for r when trying to calculate a full cone, rather than entering the true apex radius — use a full cone calculator instead if r = 0.
- Computing R^2 + r^2 and forgetting the cross term R*r in the middle of the bracket, which gives a result that can be significantly off.
- Entering diameter values instead of radius values for R and r, making the calculated volume four times too large.
Key terms
- What is a frustum?
- A frustum is the portion of a cone remaining after cutting off the top with a plane parallel to the base.
- What if the top radius is 0?
- When the top radius is 0 the frustum is a complete cone, and the formula reduces to V = (1/3) x pi x R^2 x h.
Frequently asked questions
- What is a frustum and where does it appear in real life?
- A frustum is what you get when you slice a cone parallel to its base and remove the top. Common examples include drinking cups, lampshades, buckets, and cooling towers.
- Why does the formula contain the R*r cross term?
- The full expansion of (R + r)^2 is R^2 + 2Rr + r^2, but the correct bracket is R^2 + Rr + r^2. This middle term arises from the exact integral of a linearly tapering cross-section and cannot be dropped or doubled.
- How do I find the volume if I only know the slant height, not the vertical height?
- Compute the vertical height from the slant height s using h = sqrt(s^2 - (R - r)^2), then substitute h into the frustum volume formula.