A Cone Is Exactly One Third of Its Cylinder

The surface uses the slant, not the height

The curved side of a cone is longer than the cone is tall, because it leans outwards. Its length is √(r² + h²), which for this cone is √125 = 11.18 against a height of 10. The curved area is π × r × that slant, or 175.6, and adding the circular base of 78.5 gives the 254.2 the tool reports. Using the height by mistake gives 157.1 for the side, understating it by 10.6 % — an error that shows up as too little material when a cone is cut from sheet or painted. Whenever a formula for a cone or a pyramid concerns its surface rather than its contents, check whether it wants the slant.

A heap of loose material chooses its own height

Gravel, sand and grain tipped onto flat ground do not form an arbitrary cone: the sides settle at the steepest slope the material will hold, and that angle is a property of what it is made of and how wet it is. The base radius and the height are therefore not independent — pour more of the same material and the pile widens as it grows, keeping the same slope. Estimating a stockpile from a radius and a guessed height will usually be wrong for that reason, and measuring the height directly, or the circumference and the slope, is the way to get a number worth using. Published angles vary widely between materials and with moisture, so measure the pile in front of you rather than looking one up.

Frequently asked questions

Why one third rather than a half?

Because the cone narrows in two dimensions at once as it rises, not one. At half the height the cross-section is not half the base but a quarter of it, since both the width and the depth have halved. Adding up those shrinking slices gives a third rather than the half that a shape narrowing linearly would give — a triangle, which narrows in one dimension only, is indeed half of its rectangle. The same reasoning explains why a sphere is two thirds of the cylinder that just contains it, and why the exponent in a volume formula is always one higher than in the corresponding area formula.

How do I compute a truncated cone?

Take the whole cone the shape would have been and subtract the small cone cut off the top. Extending the sides until they meet gives the full height, the full cone is computed as usual, and the removed tip is a smaller cone with the same slope. That is exactly the arithmetic behind a bucket, a plant pot and a cooling tower, and it is why doubling the depth of a tapered pot adds far more than double the soil — most of the extra volume sits in the wide part near the top.