A cone is one of the most common three-dimensional shapes you meet outside the classroom: traffic cones, party hats, ice-cream cones, conical piles of gravel, sharpened pencils, martini glasses, and the pointed roofs of towers all approximate a right circular cone. Whether you are a student checking homework, a builder estimating how much concrete a form needs, or a cook scaling a recipe for a conical mold, knowing the exact volume saves guesswork.
This cone volume calculator takes the two measurements that fully define a right circular cone — the radius of its circular base and its perpendicular height — plus a Length Unit dropdown, and instantly returns the volume together with the slant height, the base area, the lateral (curved) surface area, and the total surface area. The engine converts both radius and height to centimeters before applying the formulas, so all outputs come back in cm (cm² for areas, cm³ for volume). Adjust either number or switch the unit dropdown and every result updates, so you can feel how strongly the volume depends on the radius (it grows with the square of the radius) rather than the height (which is linear).
How to Use
- Measure or enter the Base Radius. The radius is half the diameter of the circular base. If you only have the diameter, divide it by two first.
- Measure or enter the Perpendicular Height. This is the straight vertical distance from the center of the circular base straight up to the tip (apex). It is not the slanted edge — if you measure the slanted side, use the slant-height readout instead and back-calculate.
- Pick the Length Unit from the dropdown — Centimeters (cm), Meters (m), Inches (in), or Feet (ft). Both the radius and the height are assumed to be in this unit. The engine converts them to centimeters (1 m = 100 cm, 1 in = 2.54 cm, 1 ft = 30.48 cm) before calculating.
- Read the Volume on the highlighted result card, in cubic centimeters (cm³).
- Use the supporting readouts as needed: the Slant Height (cm) is the length of the sloped side, the Base Area (cm²) is the area of the circular disk on the bottom, the Lateral Surface Area (cm²) is the area of the curved side only, and the Total Surface Area (cm²) adds the base to that.
- If your project needs a capacity in gallons or liters, the engine already gives you cm³: divide by 1,000 to get liters, or divide by 3,785 to get US gallons.
A quick sanity check: a cone holds exactly one-third of the volume of the cylinder that shares the same base radius and height. If your cone result looks like the same number as a matching cylinder, you have forgotten the one-third factor.
The Cone Volume Formula
The volume of a right circular cone is one third of the volume of the cylinder with the same base and height:
V = (1/3) × π × r² × h
Where:
- V is the volume.
- r is the radius of the circular base.
- h is the perpendicular (vertical) height.
- π is the constant pi, approximately 3.14159.
Because the radius is squared, doubling the radius quadruples the volume, while doubling the height only doubles it. That is why a short, wide funnel holds far more than a tall, thin one, even when both look impressive.
The slant height, needed for the surface area, comes straight from the Pythagorean theorem applied to the right triangle formed by the radius, the height, and the sloped side:
s = √(h² + r²)
From the slant we get the surface areas:
- Base area = π × r²
- Lateral surface area = π × r × s
- Total surface area = π × r × (r + s)
Worked example with r = 5 and h = 10: r² = 25, so V = (1/3) × π × 25 × 10 = 261.8 cubic units. The slant s = √(100 + 25) = 11.18, and the total surface area = π × 5 × (5 + 11.18) = 254.2 square units.
Why the One-Third Factor Matters
It is easy to assume a cone and a cylinder of the same height and base hold the same amount, but Archimedes himself proved the cone fills exactly one-third of that cylinder. This is not a rough rule; it follows directly from the integration of the circular cross-sections that shrink linearly from the full base radius at the bottom to zero at the apex. In practice, it means a conical storage tank that looks as big as a cylindrical drum actually holds only 33% as much — plan material, paint, or fuel for the one-third, not for the full cylinder.
If you fill a conical container to only 50% of its height, do not assume it is half full. Because the cross-sectional radius shrinks linearly, the volume below half height is just (0.5)³ = 12.5% of the total capacity — a quarter-filled cone looks visually much emptier than it really is.
Examples
Example 1 — A standard cone. With radius 5 cm and height 10 cm, the volume is 261.8 cm³, the slant height is 11.18 cm, and the total surface area is 254.2 cm². This is the built-in example and matches the classic textbook answer of about 261.8 cubic units.
Example 2 — A gravel pile. A conical pile of crushed stone has a base radius of 3 meters and a height of 7 meters. Pick Meters (m) from the unit dropdown, type 3 and 7. The engine converts to 300 cm and 700 cm and returns a volume of about 65,973,445 cm³ (≈ 66.0 m³). At a typical delivery price of about $45 per cubic meter, the pile represents roughly $2,970 of material — a useful number before you quote a customer.
Example 3 — A conical water tank. A funnel-shaped tank has radius 1,000,000 units and height 2,000 units. The engine scales with no overflow and reports a volume on the order of 2.09 × 10¹⁵ cubic units, confirming the formula handles very large magnitudes without rounding errors.
Example 4 — Scaling up. Going from radius 2, height 4 (volume 16.8) to radius 4, height 4 quadruples the radius effect: the volume jumps to 67.0, four times larger — exactly as the squared-radius term predicts.
FAQ
What is the formula for the volume of a cone?
The volume is V = (1/3) × π × r² × h, where r is the radius of the circular base and h is the perpendicular height. A cone holds exactly one-third of the volume of a cylinder with the same base radius and height.
How do I find the volume if I only know the diameter?
Divide the diameter by 2 to get the radius, then plug that radius into V = (1/3) × π × r² × h. For example, a diameter of 10 means a radius of 5, giving the same 261.8 cubic-unit result as the worked example.
Is this calculator for right circular cones or oblique cones?
This calculator is for the right circular cone, meaning the apex sits directly above the center of a circular base and the side is perfectly straight. For an oblique (tilted) cone the same volume formula still applies, but the surface-area readouts assume a right cone and will not be accurate for a tilted shape.
How do I convert cubic centimeters to gallons or liters?
One liter equals 1,000 cm³, so divide a cm³ volume by 1,000 to get liters. One US gallon equals about 3,785 cm³ (or 231 cubic inches), so divide a cubic-inch volume by 231 to get US gallons.
Why does doubling the radius quadruple the volume?
The radius appears squared in the formula, while the height appears only to the first power. If you double r, r² becomes four times larger, so the volume quadruples even with the same height. Doubling the height alone only doubles the volume.
How do I calculate the surface area of a cone?
First find the slant height s = √(h² + r²). The curved (lateral) surface area is π × r × s, and the total surface area adds the circular base: π × r × (r + s). This calculator returns both automatically.
Disclaimer
This cone volume calculator is provided for informational purposes only and does not constitute engineering, construction, or design advice. Results are based on the standard right-circular-cone formulas and the exact inputs you enter; real objects such as soil piles, funnels, and tanks may be imperfect cones. Always verify critical measurements with appropriate professionals and surveying equipment before making purchasing or structural decisions.