Conical Frustum Calculator for Volume and Area
Find the volume, slant height and surface areas of a conical frustum from two radii and height
| Property | Value |
|---|
Conical frustum calculator results show every surface measurement
Conical frustum calculator results show every surface measurement including slant height, volume, lateral area, top area, base area, total surface area and the half angle of the equivalent full cone. The table below shows example results for a frustum with a top radius of 3, a bottom radius of 6, and a height of 8.
| Property | Formula | Example Value |
|---|---|---|
| Slant height (s) | √(h² + (r1-r2)²) | 8.544 |
| Volume (V) | (1/3)πh(r1²+r2²+r1r2) | 553.581 |
| Lateral area (L) | π(r1+r2)s | 241.555 |
| Top area (T) | πr1² | 28.274 |
| Base area (B) | πr2² | 113.097 |
Conical frustum volume formula extends the cone formula
Conical frustum volume formula extends the cone formula by accounting for two radii instead of one. The formula V equals one third pi h times the sum of r1 squared, r2 squared and r1 times r2 reduces to the standard cone formula when one radius equals zero.
Why the slant height uses the radius difference
Why the slant height uses the radius difference comes from how the slanted side connects two circles of different sizes. The horizontal offset between the edges of the two circles equals the difference between r1 and r2, and combining this offset with the vertical height through the Pythagorean theorem gives the slant height.
Top area, base area and total surface area
Top area, base area and total surface area are three separate measurements that together describe the full outer surface of the frustum. Top area and base area are simply the areas of the two circular faces, calculated as pi times each radius squared, while total surface area adds these two circle areas to the lateral surface area.
Relating a frustum back to its original cone
Relating a frustum back to its original cone is possible because a frustum is always the bottom portion of some larger cone with the top sliced off. The half angle theta equals the arctangent of the radius difference divided by the height, and this angle matches the half angle of the original full cone before it was cut.
Real world examples use frustum shapes often
Real world examples use frustum shapes often because many everyday containers taper from a wider base to a narrower top or vice versa. Buckets, lampshades, drinking cups, traffic cones with flat tops and concrete forms for tapered columns are all conical frustums.
Calculating material needed for a tapered container
Calculating material needed for a tapered container starts with the volume formula to determine capacity, then uses the lateral surface area to estimate how much material is needed to wrap or coat the sloped sides of the container.
Frequently asked questions about conical frustum calculations
A conical frustum is the shape left when the top of a cone is sliced off by a cut parallel to its base. It has two circular faces of different sizes connected by a slanted lateral surface, commonly seen in buckets, lampshades and traffic cones.
The volume of a conical frustum is V equals one third times pi times height times the sum of r1 squared, r2 squared, and r1 times r2, written as V = (1/3) pi h (r1^2 + r2^2 + r1 r2). Here r1 and r2 are the two radii and h is the perpendicular height between them.
The slant height of a frustum is found by taking the square root of the height squared plus the difference between the two radii squared, written as s = square root of (h^2 + (r1 minus r2)^2). This formula comes from the Pythagorean theorem applied to the cross section of the frustum.
A full cone comes to a single point at the apex, while a frustum has two flat circular faces instead of a point because the top portion of the cone has been removed. A frustum can be thought of as the bottom section of a larger cone with the tip cut off.
The lateral surface area of a conical frustum equals pi times the sum of the two radii times the slant height, written as L = pi (r1 + r2) s. This measures only the slanted side surface and excludes the top and bottom circular faces.
If both radii are equal, the shape becomes a cylinder rather than a frustum, and the volume formula simplifies to pi times radius squared times height. The frustum formulas still work mathematically in this case and produce the same result as the cylinder volume formula.