Cone Volume Calculator
Compute every measurement of a right circular cone from two dimensions. Choose a full cone or a truncated cone (frustum), enter the base radius or diameter and height, and instantly get volume, slant height, surface areas, and apex angle. An optional cost-estimate mode lets you price material by volume.
Formula
Worked example
Full cone, radius 5, height 12: slant = sqrt(25 + 144) = sqrt(169) = 13. Volume = (1/3) * pi * 25 * 12 = 100*pi approx 314.16. Lateral area = pi*5*13 approx 204.20. Total surface area = pi*5*(5+13) approx 282.74. Half-angle = atan(5/12) approx 22.62 degrees. Frustum example, R=5, r2=2, h=8: volume = (pi/3)*8*(25+10+4) = (pi/3)*8*39 approx 326.73.
How the cone volume formula works
The volume of a right circular cone is one third of the volume of a cylinder that shares its circular base and height. A cylinder of radius r and height h has volume pi*r^2*h, so the cone built inside it holds exactly (1/3)*pi*r^2*h. This one-third relationship is not an approximation; it follows from integral calculus, where the cone is sliced into infinitely thin discs whose radii shrink linearly from r at the base to zero at the apex. The radius term is squared, which means the volume is far more sensitive to the base radius than to the height: widening the base has a much larger effect than raising the apex by the same amount. You can enter either the radius or the full diameter, whichever measurement you have to hand.
Slant height, surface area, and apex angle
The slant height is the straight-line distance along the cone's sloping side from the base edge to the apex. Because the radius, the vertical height, and the slant height form a right triangle, the slant height equals sqrt(r^2 + h^2) by the Pythagorean theorem. The curved lateral surface, which unrolls into a flat sector, has area pi*r*l, where l is the slant height. Adding the flat circular base (pi*r^2) gives the total surface area pi*r*(r+l). The half-angle at the apex, found with atan(r/h), describes the pointiness of the cone: a small angle means a sharp, narrow cone; a large angle means a wide, shallow one. The full apex angle is twice the half-angle.
Truncated cone (frustum)
A frustum is what remains after slicing a cone with a cut parallel to its base. It has two circular faces: the original base with radius R and a smaller top face with radius r2. The frustum volume formula is (pi/3)*h*(R^2 + R*r2 + r2^2), which reduces to the full-cone formula when r2 = 0. The slant height of the frustum, the distance along the sloping side between the two rim circles, is sqrt(h^2 + (R - r2)^2). Lateral surface area is pi*(R + r2)*l. Frustum shapes appear in traffic cones, drinking cups (tapered paper cups), planter pots, and the frustum sections of rocket nozzles.
Material cost estimation
Enable the cost-estimate toggle to price the fill material by volume. Enter the cost per cubic centimetre or cubic inch of the material, such as resin, concrete, powdered pigment, or ice cream mix, and the calculator multiplies that by the computed volume. This is a planning figure only; wastage, density, and mixing ratios vary by material. For bulk aggregates sold by weight, divide the volume (in cm^3 or in^3) by the material density (in the same unit) to get a mass, then multiply by the price per unit mass.
Cone and frustum examples
| Type | Base R (cm) | Top r (cm) | Height (cm) | Slant (cm) | Volume (cm³) |
|---|---|---|---|---|---|
| Full cone | 3 | - | 4 | 5.00 | 37.70 |
| Full cone | 5 | - | 12 | 13.00 | 314.16 |
| Full cone | 6 | - | 8 | 10.00 | 301.59 |
| Full cone | 10 | - | 10 | 14.14 | 1047.20 |
| Frustum | 5 | 2 | 8 | 8.54 | 326.73 |
| Frustum | 8 | 3 | 10 | 11.18 | 932.89 |
Volumes use V = (1/3)*pi*r^2*h for full cones. Slant = sqrt(r^2 + h^2). A 3-4-5 right triangle gives a clean slant height.
Frequently asked questions
Why is the cone volume one third of a cylinder?
A cone and a cylinder with the same base radius and height satisfy the exact ratio of one to three. Integral calculus shows that summing the shrinking circular cross-sections of a cone yields (1/3)*pi*r^2*h, precisely one third of the cylinder's pi*r^2*h. You can also see this physically: three conical containers of water exactly fill one cylinder of the same base and height.
What is the difference between height and slant height?
Height is the perpendicular distance from the centre of the base straight up to the apex. Slant height is the distance along the sloping outer surface from the base edge to the apex. They are related by the Pythagorean theorem: slant = sqrt(radius^2 + height^2). Volume uses the vertical height; lateral surface area uses the slant height.
What is a frustum and how is its volume different?
A frustum is a cone with its top sliced off by a cut parallel to the base, leaving two parallel circular faces. Its volume is (pi/3)*h*(R^2 + R*r + r^2), where R is the base radius, r is the top radius, and h is the height between the two faces. When r = 0, the formula reduces to the standard cone formula (1/3)*pi*R^2*h. Common frustum objects include traffic cones, tapered drinking cups, and planter pots.
How do I convert the volume to litres or gallons?
If you entered centimetres, the volume is in cubic centimetres; divide by 1000 to get litres. If you entered inches, the volume is in cubic inches; divide by 231 to get US gallons or by 277.42 for imperial (UK) gallons. Always use one consistent unit for radius and height before calculating.
What is the apex half-angle and how is it used?
The half-angle is the angle between the central axis of the cone and its sloping side, equal to atan(r/h). It describes how steep the cone is: a half-angle near 0 means a very tall, narrow cone, while a half-angle near 90 degrees means a very flat, wide cone. Engineers use the full apex angle (twice the half-angle) when designing nozzles, funnels, and tapered joints.
Can I calculate the radius from a known volume and height?
Yes. Rearranging V = (1/3)*pi*r^2*h gives r = sqrt(3V / (pi*h)). Enter the target volume and height into a separate calculation. For example, a cone with volume 314.16 cm^3 and height 12 cm has radius sqrt(3 * 314.16 / (pi * 12)) = sqrt(25) = 5 cm.