A conical frustum is a cone with its tip cut off parallel to the base: two circles of different sizes joined by a sloping wall. Buckets, flowerpots, paper cups and lamp shades all have this shape. Give this calculator both radii plus the vertical height, the slant height or the volume, and it returns everything else, including capacity in liters and US gallons and the height of the cone the frustum came from.
How to use the conical frustum calculator
- In I know the, choose Both radii and the height, Both radii and the slant height or Both radii and the volume (find the height).
- Enter the Bottom radius (R) and Top radius (r). Measure the diameter across each rim and halve it. A top radius of 0 turns the shape into a full cone.
- Enter the Height (h), Slant height (s) or Volume (V), in the same unit, and pick the Units. Volume is in cubic units, such as in³ when the unit is inches.
- Read the volume, or the height in volume mode. The list adds the slant height, capacity in liters and US gallons, the lateral, bottom, top and total surface areas, the side slope from the base and the height of the full cone.
Conical frustum formulas
Where the volume formula comes from
Extend the sloping wall until it closes to a point. The frustum is then a large cone minus the small cone that was sliced off. Similar triangles fix the large cone’s height: with R the larger radius, H = hR ÷ (R − r). Subtracting ⅓πr2(H − h) from ⅓πR2H leaves a factor R3 − r3 = (R − r)(R2 + Rr + r2), and the (R − r) cancels. That middle Rr term is exactly what quick averaging misses.
Worked example: potting soil for a tapered planter
A round planter measures 12 in across the bottom and 16 in across the top inside, and it is 14 in deep. Enter R = 6, r = 8 and h = 14 in inches.
- Volume: ⅓ × π × 14 × (36 + 48 + 64) ≈ 2,169.79 in³
- Capacity: about 35.56 L or 9.39 US gal; divide by 1,728 for about 1.26 ft³ of potting mix
- Slant height: √(2² + 14²) ≈ 14.14 in, so the inside wall covers about 622.00 in² if you plan to seal or line it
- Side slope from base: 98.13°, meaning the wall leans outward 8.13° from vertical
- Height of the full cone: 14 × 8 ÷ 2 = 56 in
Cutting a lamp shade or cone pattern
A frustum’s wall unrolls into a curved band, a slice of a ring. To cut it from fabric, paper or sheet metal you need the slant length of the whole cone, not just the frustum. With slant height s and R the larger radius, the outer pattern radius is s × R ÷ (R − r), the inner radius is that minus s, and the band spans 360 × R ÷ (outer radius) degrees.
Take a shade 16 in across the bottom and 10 in across the top, measuring 11 in along the side. Choose Both radii and the slant height with R = 8, r = 5 and s = 11. The shade is 10.58 in tall and needs 449.25 in² of fabric, about 3.12 ft² before seam allowance. The pattern’s outer radius is 11 × 8 ÷ 3 ≈ 29.33 in, the inner radius ≈ 18.33 in and the angle ≈ 98.2°. As a check, the outer arc must equal the bottom circumference, 2π × 8 ≈ 50.27 in, which the sector calculator can confirm.
Common mistakes
- Diameters in the radius boxes. That doubles every length and multiplies the volume by eight.
- Averaging. A cylinder with the average radius undercounts by exactly πh(R − r)2 ÷ 12, while averaging the end areas overcounts by twice that. For the gently tapered planter the errors are under 1.5%. For a steep frustum with radii of 6 and 2 and a height of 10 they grow to −7.7% and +15.4%.
- Slant height entered as height. A tape run along the outside wall gives the slant height. Choose that option instead of typing it into the height box.
- Outside instead of inside dimensions. For capacity, measure inside the walls.
For straight walls use the cylinder calculator; for a whole cone or a square hopper, see the cone and pyramid frustum calculators.
Frequently asked questions
What is the formula for the volume of a conical frustum?
V = ⅓πh(R² + Rr + r²), where R and r are the radii of the two circular ends and h is the perpendicular height. A frustum with radii of 5 and 3 and a height of 6 holds ⅓ × π × 6 × 49 ≈ 307.88 cubic units.
Does it matter which end I call the bottom?
Not for the volume, slant height or surface area, because the formulas treat the two radii the same way. For a bucket or flowerpot that is wider at the top, enter the larger radius as the top radius. Only the side slope changes: it reads more than 90° because the wall leans outward.
How do I find the height of a frustum from its slant height?
The height, the slant height and the difference between the radii form a right triangle, so h = √(s² − (R − r)²). Radii of 8 in and 5 in with an 11 in slant height give h = √(121 − 9) ≈ 10.58 in.
Can I just average the two radii?
Only for a rough guess. A cylinder with the average radius always comes out too small, and averaging the two end areas always comes out too large. For a steep frustum with radii of 6 and 2 and a height of 10, the errors are about −7.7% and +15.4%.
What is the height of the full cone?
It is the height of the complete cone you would get by extending the sloping side until it meets at a point. By similar triangles it equals h × (larger radius) ÷ (difference of the radii), so a frustum 6 units tall with radii of 5 and 3 was cut from a cone 15 units tall.