Frustum of a Cone Calculator
Enter your dimensions below to calculate volume, slant height, lateral surface area, and total surface area.
What is a frustum?
A frustum is the portion of a cone (or pyramid) left after cutting the top off with a plane parallel to the base. The shape looks like a cone with the tip removed. In real life, frustums appear in cups, buckets, hoppers, nozzles, lampshades, and many engineering components.
This page focuses on a conical frustum, which is defined by:
- Bottom radius (R)
- Top radius (r)
- Height (h) measured vertically
Formulas used by this frustum calculator
1) Volume
V = (πh/3) × (R² + Rr + r²)
2) Slant height
s = √((R − r)² + h²)
3) Lateral surface area
Alateral = π(R + r)s
4) Total surface area
Atotal = Alateral + πR² + πr²
How to use the calculator
- Enter the bottom radius R.
- Enter the top radius r.
- Enter the vertical height h.
- (Optional) add a unit label like cm or in.
- Click Calculate to see all results instantly.
Example calculation
Suppose a frustum has R = 8, r = 5, and h = 12:
- Volume = (π × 12 / 3) × (64 + 40 + 25) = 516π ≈ 1621.061
- Slant height = √((8 − 5)² + 12²) = √153 ≈ 12.369
- Lateral area = π(8 + 5)(12.369) ≈ 505.159
- Total area = Lateral + Top + Bottom ≈ 505.159 + 78.540 + 201.062 = 784.761
Where frustum math is useful
- Estimating material needs for containers and molds
- Calculating liquid capacity in tapered tanks
- Designing industrial funnels and transition ducts
- 3D modeling and CAD geometry checks
- Architecture and product design prototyping
Common mistakes to avoid
- Mixing diameter and radius values (radius is half of diameter)
- Using inconsistent units (for example, inches for radius and centimeters for height)
- Confusing slant height with vertical height
- Forgetting that area units are squared and volume units are cubed
Quick FAQ
Can the top radius be zero?
Yes. If r = 0, the frustum becomes a full cone.
Can top and bottom radii be equal?
Yes. If R = r, the shape behaves like a cylinder for volume and surface calculations.
Do I need to enter units?
No. Units are optional, but adding them helps label your results clearly.