Find the volume of a cone.
The same base-times-height calculation as a cylinder, but exactly one-third of it.
Volume: (1/3) × π × radius² × height
A cone has a circular base that narrows to a single point, the apex, like an ice-cream cone, a birthday cap, a funnel, a tent or a heap of grain on a threshing floor. The cone volume calculator takes the base radius and the vertical height and returns (1/3)πr²h.
The one-third is the key fact: a cone holds exactly a third of what a cylinder with the same base and height would hold. That makes the formula useful for estimating the volume of a pile of sand, gravel or wheat, the capacity of a conical hopper or funnel, or the ice cream in a cone. Cones appear in Class 9 and 10 mensuration, often combined with cylinders and hemispheres in composite shapes.
1. Measure the radius r of the circular base. For a heap, measure the circumference of its base and use r = C ÷ (2π).
2. Measure the vertical height h from the centre of the base straight up to the apex.
3. If you know only the slant height l, find the vertical height with h = √(l² − r²).
4. Compute the base area πr².
5. Multiply by h and divide by 3: V = (1/3)πr²h.
6. Express the result in cubic units, or in litres if needed.
Slice a cone into thin discs parallel to its base. A disc at a fraction t of the way down from the apex has radius tr, so its area is πr²t². Adding these areas over the height is the same as integrating πr²t² h dt from 0 to 1, which gives πr²h × 1/3. The t² inside is what produces the one-third. You can also show it by experiment: three conefuls of water exactly fill a cylinder of the same base and height.
A cone has two heights. The vertical height h is used for volume. The slant height l, measured along the sloping side from the rim to the apex, is used for the curved surface area, πrl. They are linked by Pythagoras: l² = r² + h². A common error is to put the slant height into the volume formula, which always overestimates, because the slant height is longer than the vertical height.
The one-third rule is not special to cones. Any pyramid has volume (1/3) × base area × height, and a cone is a pyramid with a circular base. If the top of a cone is cut off, the remaining frustum, like a bucket, has volume (1/3)πh(R² + Rr + r²). A pile of loose material such as sand forms a cone whose slope depends on the angle of repose, so measuring the base circumference and height gives a quick estimate of its volume.
A farmer in Punjab has a conical heap of harvested wheat with a base radius of 1.8 metres and a vertical height of 1.2 metres.
Answer: Volume 4.0715
Using the slant height instead of the vertical height.
Forgetting to divide by 3, which gives a cylinder's volume.
Entering the diameter instead of the radius, which quadruples the answer.
Measuring the height of a heap along its surface rather than straight up from the ground at the centre.
Treating a bucket-shaped frustum as a full cone, which overestimates the volume.
Estimating the volume of heaps of grain, sand, gravel or coal.
Finding the capacity of conical funnels, hoppers and silos with conical bottoms.
Working out the ice cream or filling in cones for food businesses.
Designing tents, lampshades and traffic cones.
Solving composite-solid problems in Class 9 and 10.
Why is a cone exactly a third of the equivalent cylinder?
A cone narrows evenly down to a point. It can be shown, using geometry and calculus, that this tapering shape holds exactly one-third the volume of a cylinder with the same base and height.