Frustum Volume Calculator
Find the volume of a conical frustum.
Results update as you type.
How it works
A frustum is what remains when the top of a cone is cut off by a plane parallel to the base, a shape seen in buckets, lampshades, drinking cups, and many containers, and this calculator finds its volume and surface. You enter the bottom radius, the top radius, and the height, and it returns the volume, the lateral surface area, and the slant height. The volume formula multiplies pi times a third of the height by the combined term of the two radii squared plus their product, which smoothly interpolates between a full cone, when the top radius is zero, and a cylinder, when both radii are equal. The slant height, the distance along the sloped side, is found with the Pythagorean theorem from the height and the difference between the radii, and the lateral surface area, the curved side without the two circular ends, uses that slant. Frustums are common in engineering and everyday objects, so being able to compute their volume is useful for finding the capacity of a bucket or cup, the material in a lampshade or funnel, or quantities in construction such as concrete for a tapered pier. The two radii must be non-negative and the height positive, with at least one radius greater than zero. Enter the two radii and the height to compute the frustum's volume and surface.
V = (πh/3)(R² + Rr + r²)
A frustum is a cone with its top sliced off parallel to the base. Its volume is pi times the height divided by 3, times the sum of the squares of the two radii plus their product. The slant height comes from the Pythagorean theorem on the height and the difference of radii, and the lateral surface uses that slant.
Source:Solid geometry
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter bottom radius (r), top radius (r) and height (h).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Volume updates instantly as you type — no button to press.
Frequently asked questions
What is a frustum?
A frustum is the portion of a cone (or pyramid) that remains after the top is cut off by a plane parallel to the base. A conical frustum has two circular faces of different radii connected by a slanted side. Everyday examples include buckets, drinking cups, lampshades, and funnels, all of which taper from a wider base to a narrower top.
How do you calculate a frustum's volume?
Multiply pi times the height divided by three by the sum of the two radii squared plus their product. This formula bridges the cone and cylinder: if the top radius is zero it becomes the cone volume, and if the two radii are equal it becomes the cylinder volume. The calculator applies it directly from your two radii and height.
What is the slant height?
The slant height is the distance measured along the sloped side of the frustum, from the edge of the bottom circle to the edge of the top circle. It is found with the Pythagorean theorem using the vertical height and the difference between the two radii. The slant height is needed to compute the lateral surface area.
What is the lateral surface area?
The lateral surface area is the area of the curved, slanted side of the frustum, not including the two circular ends. It equals pi times the sum of the two radii times the slant height. This is useful for finding how much material covers the sloped side, such as the fabric of a lampshade or the metal of a funnel.
Where are frustum calculations useful?
They help find the capacity of tapered containers like buckets and cups, the material needed for lampshades, funnels, and ducts, and quantities in construction such as concrete for a tapered column or the volume of a pile of material. Any object that narrows from a wider circular base to a smaller circular top is a frustum.