How to Calculate Frustum
What is Frustum?
A frustum is a truncated cone — a cone with the top cut off by a plane parallel to the base. It appears in architecture (towers, buckets) and engineering (funnels, lampshades).
Formula
V = (πh/3)(R² + r² + Rr); l = √(h² + (R−r)²)
- R
- base radius (large) (length)
- r
- top radius (small) (length)
- h
- height (length)
- l
- slant height (length)
- V
- volume (length³)
Step-by-Step Guide
- 1Volume = (πh/3)(R² + r² + Rr)
- 2Slant height l = √(h² + (R−r)²)
- 3Lateral surface = π(R+r)l
- 4Total surface = π[l(R+r) + R² + r²]
Worked Examples
Input
R=5, r=3, h=8
Result
Volume ≈ 410.5, Lateral SA ≈ 201.1
Input
R=10, r=5, h=12
Result
Volume ≈ 2199.1
Frequently Asked Questions
What happens to the frustum formula if r = 0?
If r = 0, you get a cone: V = (πh/3)R², which makes sense geometrically.
Is the slant height the same as the lateral surface area edge?
Yes, the slant height l is the length along the slanted edge of the frustum surface.
How is a frustum created?
Cut a cone with a plane parallel to the base, removing the top point. What remains is the frustum.
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