Frustum of a Cone Calculator
A frustum is the solid left when a cone is cut by a plane parallel to its base, creating a shape with two circular faces of different sizes. Enter the bottom radius, top radius, and height below to get the total surface area, lateral (side) surface area, both base areas, volume, and slant height. The step-by-step panel shows every calculation with your numbers substituted in.
What is a frustum of a cone?
A frustum (sometimes spelled frustrum) is the portion of a solid - in this case a right circular cone - that remains after cutting off the top with a plane parallel to the base. The result is a solid bounded by two parallel circular faces (called bases) of different radii and a sloping lateral surface connecting them. Frustums appear constantly in engineering and everyday objects: buckets, drinking cups, cooling towers, lampshades, funnels, and the stages of a rocket nozzle are all conical frustums. Computing the surface area tells you how much material is needed to make the shell, while the volume tells you how much the object can hold.
How the formulas work
Let R be the bottom (larger) radius, r the top (smaller) radius, and h the perpendicular height. The first step is always finding the slant height s = sqrt(h^2 + (R - r)^2), which is the straight-line distance along the slope from the edge of the bottom circle to the edge of the top circle. Think of unrolling the lateral surface: it becomes a flat ring (an annular sector), and its area is exactly pi * (R + r) * s. Add the two flat circular ends - pi * R^2 for the bottom and pi * r^2 for the top - to get the total surface area. The volume formula V = (pi/3) * h * (R^2 + r^2 + R*r) follows from subtracting the small cone that was removed from the full cone.
Worked example step by step
Suppose R = 6 cm, r = 3 cm, h = 8 cm. Step 1: slant height s = sqrt(8^2 + (6-3)^2) = sqrt(64 + 9) = sqrt(73) = 8.544 cm. Step 2: lateral area = pi * (6 + 3) * 8.544 = pi * 9 * 8.544 = 241.5 cm^2. Step 3: bottom base = pi * 6^2 = 113.1 cm^2. Step 4: top base = pi * 3^2 = 28.3 cm^2. Step 5: total surface area = 241.5 + 113.1 + 28.3 = 382.9 cm^2. Step 6: volume = (pi/3) * 8 * (36 + 9 + 18) = (pi/3) * 8 * 63 = 527.8 cm^3.
Metric vs. imperial units
The formulas are identical in metric and imperial units - only the labels change. In metric the most common length unit for geometry problems is centimetres, giving areas in cm^2 and volumes in cm^3. In imperial the standard is inches, giving in^2 and in^3. If your measurements are in different units (for example, height in feet but radii in inches), convert everything to the same unit before entering the values. One foot is 12 inches; one metre is 100 centimetres. This calculator uses the same unit for all three dimensions, so the conversion step only needs to happen once.
Frustum surface area and volume formulas
| Component | Formula | What it measures |
|---|---|---|
| Slant height | s = sqrt(h^2 + (R - r)^2) | Length along the sloped side |
| Lateral surface area | A_lat = pi * (R + r) * s | Area of the sloped band |
| Bottom base area | A_base = pi * R^2 | Area of the larger circle |
| Top base area | A_top = pi * r^2 | Area of the smaller circle |
| Total surface area | A = A_lat + A_base + A_top | Full outer surface |
| Volume | V = (pi/3) * h * (R^2 + r^2 + R*r) | Enclosed interior space |
All variables: R = bottom radius, r = top radius, h = height, s = slant height = sqrt(h^2 + (R-r)^2).
Frequently asked questions
What is the difference between lateral surface area and total surface area?
Lateral surface area is the area of the sloped side of the frustum only - it excludes both circular bases. Total surface area is the sum of the lateral area and the two base circles (the larger bottom base and the smaller top base). If you are calculating how much sheet metal you need to wrap around the outside of a bucket without the top or bottom lids, use only the lateral area. If you need to enclose the solid entirely, use the total surface area.
What happens when the top radius is zero?
When r = 0, the frustum becomes a complete cone. The formula still works perfectly: the top base area becomes pi * 0^2 = 0, and the slant height becomes sqrt(h^2 + R^2), which is exactly the slant height of a right circular cone. You can verify this against a standard cone surface area calculator.
What happens when the top and bottom radii are equal?
When R = r, the frustum becomes a cylinder. The slant height simplifies to just the height (sqrt(h^2 + 0) = h), the lateral area becomes the familiar 2*pi*R*h, and the volume becomes pi*R^2*h. Again the general frustum formula reduces correctly to the cylinder formula.
How do I find the slant height if I already know it?
The slant height s is calculated from the other three inputs: s = sqrt(h^2 + (R - r)^2). If you physically measured the slope length of a real object, you can verify it matches this formula. If your given dimension is the slant height rather than the vertical height, rearrange to find h = sqrt(s^2 - (R - r)^2) and enter that as the height.
Can I use this calculator for a truncated pyramid?
No - a truncated pyramid (frustum of a pyramid) has rectangular or polygonal bases, not circular ones. The formulas are different. This calculator applies only to a right circular cone frustum, where both bases are circles centered on the same vertical axis.
How is the volume of a frustum derived?
Imagine extending the frustum back into a complete cone of height H_big. The volume of that full cone is (pi/3)*H_big*R^2. The removed top cone has volume (pi/3)*H_small*r^2. Subtract the two and simplify using the similar-triangles relationship between H_big, H_small, R, and r, and you arrive at V = (pi/3)*h*(R^2 + r^2 + R*r), where h is the frustum height. This elegant result is sometimes called the prismatoid formula.