Frustum Volume Calculator
Enter the bottom radius, top radius and height of a frustum — a cone with the tip cut off — to get its volume and surface area, drawn to scale. Click any diameter or the height on the drawing to change it.
Dimensions
FrustumResult
Volume of a frustum
A frustum is a cone with its top sliced off parallel to the base. Its volume blends the two radii: V = ⅓·πh(R² + Rr + r²). When the top radius equals the bottom it becomes a cylinder; when the top radius reaches zero it becomes a full cone.
Editing on the drawing
The bottom diameter, top diameter and height are each editable — click a value, type a new one and press Enter to redraw the frustum to scale.
How to find the volume of a frustum
A frustum is what is left when you slice the top off a cone with a cut parallel to its base — the shape of a bucket, a lampshade or a cooling tower. Three measurements set a conical frustum: the bottom radius, the top radius and the height.
The formulas
The volume blends the two radii through the term R² + Rr + r² rather than simply averaging them. The slant height runs up the sloping side between the two rims.
Worked example
Take a frustum with bottom radius 50, top radius 28 and height 78.
From the radius difference and the height.
Blend the two radii.
Two circular ends plus the sloping band.
What else the calculator returns
You also get the lateral (sloping band) area, the slant height and both diameters. Set the top radius equal to the bottom and the frustum becomes a cylinder; set the top radius to zero and it becomes a full cone — the formula handles both limits.
Where it comes up
Frustum volumes describe buckets and tapered tanks, lampshades, drinking cups, hoppers, cooling towers and the transition pieces in ductwork.
Frequently asked questions
What is a frustum?
It is the part of a cone or pyramid left between the base and a cut made parallel to that base — in other words, the solid with the tip removed.
How do you find the volume of a frustum?
Use both radii and the height: V = ⅓·πh(R² + Rr + r²).
Why can't I just average the two radii?
Averaging the radii underestimates the volume. The correct blend, R² + Rr + r², comes from integrating the cone's changing cross-section between the two cuts.
What happens when the top radius equals the bottom?
The frustum becomes a cylinder and the formula reduces to πR²h. When the top radius is zero, it becomes a full cone, ⅓πR²h.
Is the slant height the same as the height?
No. The height is vertical, while the slant height, √((R − r)² + h²), runs along the sloping side and is longer.