Cone Volume Calculator
Volume, surface area, and slant height of a cone from radius and height.
What it does
Cones hide a tidy fact: any cone holds exactly one third of the cylinder built on the same base and height. A party hat with radius 5 cm and height 12 cm therefore holds a third of pi x 25 x 12, which is 314.16 cm3 or 0.31 L. The tool above runs that with surface and slant beside it.
The metric default works the classic 5-12-13 triangle. Slant height is sqrt(25 + 144) = 13 cm, side area is pi x 5 x 13, and adding the pi x 25 base lands surface at 282.7 cm2. Imperial mode mirrors the idea at radius 2 in and height 5 in for 0.09 gal.
Two references agree on the setup. Lumen prealgebra states volume as one third pi r squared h and works radius 2 with height 6 to 25.12 cubic inches, while Khan Academy walks the same formula and adds surface as pi r squared plus pi r times slant. Full pi here lands a hair above their 3.14 roundings.
Radius versus diameter trips nearly everyone once. The formula wants radius, so halve any across measurement first; feeding diameter straight in quadruples the answer. Height means the straight drop from tip to base center, never the slant edge.
How to use
- Pick metric (cm) or imperial (inches).
- Type the base radius, halving diameter first.
- Type the straight height from base to tip.
- Read volume first in liters or gallons.
- Read surface for material and slant for layout.
Formula
- r
- Base radius, not diameter.
- h
- Perpendicular height from base to tip.
- s
- Slant height along the side.
Right circular cones only; funnels with bent sides need other treatment. Pi runs at full precision, so expect cents above 3.14 textbook lines.
Worked example
Example: radius 5 cm, height 12 cm
Metric defaults: radius 5 cm, height 12 cm.
- Square: 5 x 5 = 25. Times height: 25 x 12 = 300.
- Volume: pi x 300 / 3 = 314.16 cm3 = 0.31 L.
- Slant: sqrt(25 + 144) = 13 cm.
- Surface: pi x 5 x 13 + pi x 25 = 282.7 cm2.
- Tool shows 0.31 L, 282.7 cm2, 13 cm.
Reference table
| Dims | Volume | Surface |
|---|---|---|
| 5 cm x 12 cm | 0.31 L | 282.7 cm2 |
| 10 cm x 10 cm | 1.05 L | 758.4 cm2 |
| 2 in x 5 in | 0.09 gal | 46.4 in2 |
| 3 in x 4 in | 0.16 gal | 75.4 in2 |
Limits
- Right cones only; oblique tips shift volume.
- Hollow walls ignored, so shells hold slightly less.
- Radius range caps keep floats exact.
- Gallons are US liquid gallons.
Common mistakes
- Typing diameter where radius belongs.
- Using slant height as h in the volume.
- Forgetting the one third factor.
- Mixing cm inputs with gallon hopes.
- Skipping the base in surface totals.
Try it yourself
Quick checks. Answers sit tucked under each question.
Radius 10 cm, height 10 cm?
pi x 100 x 10 / 3 = 1047.2 cm3 = 1.05 L.
Radius 3 in, height 4 in?
37.70 in3 = 0.16 gal, surface 75.4 in2, slant 5 in.
Radius 6 cm, height 8 cm?
301.59 cm3 = 0.3 L, surface 301.6 cm2, slant 10 cm.
Questions
Why one third?
A cone fills exactly a third of its matching cylinder.
Volume of 5 by 12 cm?
0.31 L with 282.7 cm2 of surface.
Slant or height for volume?
Height. Slant only serves surface math.
Is surface with or without base?
With base here: side plus pi r squared.
What about frustums?
Truncated cones need top and bottom radii, outside this page.
Why 0.31 not 0.314?
Two decimal display; full value sits behind it.