Cone Volume Formula
A cone is a three-dimensional solid with a circular base that tapers to a single point, called the apex. The volume depends on both the base area and the vertical height from base to apex.
For a right cone (where the apex sits directly above the center), multiply the circular base area by the height and divide by three. For a truncated cone (frustum), the formula accounts for both the bottom and top circular openings.
V = (1/3) × π × r² × h
V_frustum = (1/3) × π × h × (R² + R×r + r²)
V— Volume of the coner— Radius of the cone base (or top radius for a frustum)h— Perpendicular height from base to apexR— Radius of the base (bottom opening of a frustum)V_frustum— Volume of a truncated cone
Calculating Cone Volume Step by Step
To find the volume of a cone, gather two measurements: the radius of the circular base and the perpendicular height from that base to the apex.
- Measure or identify the base radius — This is the distance from the center of the circular base to its edge. If you have the diameter, divide by two.
- Measure the height — This must be the perpendicular distance, not the slant height. Use a slant height calculator if you only know the angle and slant length.
- Substitute into the formula — Multiply π by the radius squared, then by the height, then divide the result by three.
- Check your units — The answer will be in cubic units. If height is in inches and radius in inches, volume is in cubic inches.
For practical examples: a cone with radius 3 cm and height 4 cm holds about 37.7 cm³. A cone with radius 1 inch and height 6 inches contains roughly 6.3 cubic inches.
Truncated Cones and Frustums
A truncated cone (or frustum) is created when a horizontal slice removes the top portion of a cone. Common examples include flowerpots, lampshades, buckets, and truncated conical champagne glasses.
Instead of subtracting the small cut-off cone from the large original cone, use the direct frustum formula:
V = (1/3) × π × h × (R² + R×r + r²)
where R is the larger base radius, r is the smaller top radius, and h is the height between them. A standard terra cotta pot with a 10 cm bottom opening, 8 cm top opening, and 12 cm height contains approximately 1,030 cm³ of soil.
Right Cones vs. Oblique Cones
A right cone has its apex positioned directly above the center of the base, creating a symmetrical shape. An oblique cone leans to one side, with the apex offset from the base's center, much like a tilted cylinder.
A crucial geometric principle: the volume formula remains identical for both. As long as you use the perpendicular height (the shortest distance from base to apex, not along a slanted edge) and the base radius, the calculation produces the correct result. This works because volume depends on base area and perpendicular height, not the cone's orientation.
Common Pitfalls and Practical Tips
Avoid these mistakes when calculating cone volumes:
- Confusing height with slant height — Height is always the perpendicular distance from base center to apex. Slant height runs along the cone's surface. Using slant height in the formula produces incorrect results. If only slant height is known, apply the Pythagorean theorem: h² + r² = (slant height)².
- Forgetting the one-third factor — Many people mistakenly calculate (π × r² × h) without dividing by three. This common error produces a volume one-third larger than the true value. Always include the 1/3 multiplier.
- Mixing radius and diameter — Ensure you're using radius, not diameter. If your measurement is diameter, divide it by two first. Squaring the diameter instead of the radius throws off the result by a factor of four.
- Ignoring unit consistency — If height is in centimeters and radius in inches, convert to matching units before calculating. Volume units depend on input units—cubic inches from inches, cubic centimeters from centimeters.