Cone Volume Calculator
Find the volume of a cone from its radius or diameter and height, plus the slant height, surface area, liters and gallons.
The volume of a cone is one third of pi times the radius squared times the height. A cone with a 3 cm radius and a 4 cm height holds 37.7 cubic cm, one third of the 113.1 cubic cm a cylinder of the same size holds. Its slant height is 5 cm and its total surface area 75.4 square cm.
Results are for a perfect right circular cone, with the tip directly above the center of the base, rounded to two decimals. Real cones and piles are rarely perfect, and the calculator does not account for wall thickness, so measure the inside of a container if you want the volume it holds.
Volume
37.7 cubic cm
- Slant height
- 5 cm
- Side (lateral) area
- 47.12 square cm
- Base area
- 28.27 square cm
- Total surface area
- 75.4 square cm
- In liters
- 0.04 L
- In US gallons
- 0.01 gal
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About the Cone Volume Calculator
A cone holds exactly one third of the cylinder that shares its base and height, a result Euclid proved in Book XII of the Elements. That is where the one third in V = 1/3 x pi x r squared x h comes from. For a 3 cm radius and a 4 cm height, the base circle is pi x 9, about 28.27 square cm; times the height that is 113.1 cubic cm for the cylinder, and a third of it, 37.7 cubic cm, for the cone. The height here is the straight-up height from the center of the base to the tip, not the length along the side. That sloping length is the slant height, the hypotenuse of a right triangle made by the radius and the height: for 3 and 4 it is 5. The slant height sets the side area, pi x r x slant, 47.12 square cm here, and adding the base circle gives the total surface area of 75.4 square cm. If you measured across the base, choose diameter and the calculator halves it: a 10 cm diameter with a 9 cm height gives 235.62 cubic cm. A conical pile 2 ft in radius and 6 ft tall holds 25.13 cubic feet, about 188.01 US gallons.
Frequently asked questions
What is the formula for the volume of a cone?+
V = 1/3 x pi x r squared x h, where r is the base radius and h the height. A cone with a 5 in radius and a 12 in height holds 314.16 cubic inches, about 1.36 US gallons.
How do I find the slant height of a cone?+
Use the Pythagorean theorem on the radius and the height: slant = the square root of r squared plus h squared. A 5 in radius and 12 in height give a slant height of 13 in.
What is the surface area of a cone?+
The side, or lateral, area is pi x r x slant height, and the total adds the base circle, pi x r squared. For a 5 in radius and a 12 in height the total is 282.74 square inches. An open cone, like a paper cup, has only the side area.
Why is a cone one third of a cylinder?+
Euclid proved that any cone is one third of the cylinder with the same base and the same height. So, ideally, three cones of water poured in fill the matching cylinder to the brim.
Formulas: Weisstein EW, "Cone," MathWorld; Euclid, Elements, Book XII, Proposition 10. Units: NIST Handbook 44, Appendix C.
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