Cone Frustum: worked example
| Bottom radius (R) | 6 |
|---|---|
| Top radius (r) | 3 |
| Height (h) | 8 |
| Volume | 527.8 |
| Slant height | 8.5 |
| Total surface area | 382.9 |
Cone Frustum: formula
Enter the bottom radius, the top radius and the vertical height of the truncated cone, in the same unit. A frustum is a cone with the tip cut off by a plane parallel to the base. The volume uses pi times the height times the sum of R squared, R times r and r squared, divided by three.
The slant height comes from the Pythagorean theorem with the height and the difference of radii.
Cone Frustum: changing Bottom radius (R)
| Bottom radius (R) | Volume | Slant height | Total surface area |
|---|---|---|---|
| 4.8 | 389.1 | 8.2 | 301.6 |
| 5.4 | 455.4 | 8.4 | 340.3 |
| 6 (example) | 527.8 | 8.5 | 382.9 |
| 6.6 | 606.2 | 8.8 | 429.7 |
| 7.2 | 690.6 | 9 | 480.7 |
Cone Frustum: compare input ranges
| Input | Tested from | To | Volume: low input | Volume: high input | Swing |
|---|---|---|---|---|---|
| Bottom radius (R) | 5.4 | 6.6 | 455.4 | 606.2 | 150.8 (29%) |
| Height (h) | 7 | 9 | 461.8 | 593.8 | 131.9 (25%) |
| Top radius (r) | 2.7 | 3.3 | 498.4 | 558.7 | 60.3 (11%) |
Cone Frustum: individual input comparisons
| Input | Value | Volume | Slant height | Total surface area |
|---|---|---|---|---|
| Bottom radius (R) | 4.8 | 389.1 | 8.2 | 301.6 |
| 5.4 | 455.4 | 8.4 | 340.3 | |
| 6 | 527.8 | 8.5 | 382.9 | |
| 6.6 | 606.2 | 8.8 | 429.7 | |
| 7.2 | 690.6 | 9 | 480.7 | |
| Top radius (r) | 2.4 | 470.5 | 8.8 | 362.7 |
| 2.7 | 498.4 | 8.7 | 372.5 | |
| 3 | 527.8 | 8.5 | 382.9 | |
| 3.3 | 558.7 | 8.4 | 394 | |
| 3.6 | 591.1 | 8.4 | 405.7 | |
| Height (h) | 6 | 395.8 | 6.7 | 331 |
| 7 | 461.8 | 7.6 | 356.7 | |
| 8 | 527.8 | 8.5 | 382.9 | |
| 9 | 593.8 | 9.5 | 409.6 | |
| 10 | 659.7 | 10.4 | 436.6 |
Cone Frustum: combined input changes
| Bottom radius (R) / Top radius (r) | 2.4 | 3 | 3.6 |
|---|---|---|---|
| 4.8 | 337.8 | 389.1 | 446.4 |
| 5.4 | 401.1 | 455.4 | 515.7 |
| 6 | 470.5 | 527.8 | 591.1 |
| 6.6 | 545.9 | 606.2 | 672.6 |
| 7.2 | 627.3 | 690.6 | 760 |
Cone Frustum: which assumptions move the result?
Cone Frustum: values and definitions
| Quantity and definition | Value used |
|---|---|
| Bottom radius (R) | 6 |
| Top radius (r) | 3 |
| Height (h) | 8 |
| Volume | 527.8 |
| Slant height | 8.5 |
| Total surface area | 382.9 |
Cone Frustum: half and double Bottom radius (R)
| Bottom radius (R) | Volume |
|---|---|
| 3 | 226.2 |
| 6 | 527.8 |
| 12 | 1,583.4 |
Common error: Averaging the two radii and using a cylinder formula. That works only for a very slight taper, and it misses the volume as the sides lean in.
The top and bottom must be parallel circles with a common axis. Inside dimensions apply to capacity, and outside dimensions apply to coatings. Sloped walls make the slant height, not the vertical height, the length of the side.
Rounding in the slant affects surface area more than volume.
Geometry calculators: Capsule Volume Calculator, Triangular Prism Volume Calculator, Law of Sines Calculator, Cube Calculator, Equilateral Triangle Calculator, Trigonometry Calculator.
A frustum is a cone with the tip sliced off, so its volume is a difference of two cones folded into one formula. A bucket with a bottom radius of 6, a top radius of 3 and a height of 8 holds about 528 cubic units. The slant height, about 8.54, is the length of the sloping wall, and it is longer than the straight-up height whenever the radii differ.
Use the perpendicular height between the two circles, not the slanted side. The slanted side belongs only in the surface area.
See the cone calculator for the full shape, the cylinder and pyramid pages for other solids, and the annulus calculator for ring areas. The Pythagorean theorem page helps with the slant, and the unit conversion and aquarium calculators convert volume to gallons.
What is a frustum?
The part of a cone or pyramid that remains after the top is cut off by a plane parallel to the base.
How do I find the slant height of a frustum?
Take the square root of the height squared plus the square of the difference between the bottom and top radii, which is the Pythagorean theorem applied to the side.
Where do frustums appear?
In buckets, cups, lampshades, hoppers, chimneys and many containers with sloping sides, wherever a cone shape is cut off before its tip.