Math

Cone Volume Calculator

Calculate the volume, slant height, lateral surface area, base area, and total surface area of a cone from radius and height.

Enter radius and height to see the result.

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Cone Formulas

All cone measurements derive from the base radius (r) and perpendicular height (h).

Volume = (1/3) × π × r² × h

Slant height (l) = √(r² + h²)

Lateral SA = π × r × l

Total SA = π × r × (l + r)

Why a Cone is 1/3 of a Cylinder

One of the most elegant results in solid geometry is that the volume of a cone is exactly one-third of the volume of a cylinder with the same base radius and height. This can be demonstrated physically by filling a cone with water and pouring it into the corresponding cylinder — you need to do this exactly three times to fill the cylinder.

The formal proof uses integral calculus. The cylinder can be sliced into thin horizontal discs. At height y from the base, a cone has radius r × (1 − y/h), giving a disc area of π × r² × (1 − y/h)². Integrating this from 0 to h and dividing by the cylinder volume π × r² × h gives exactly 1/3.

This relationship extends to other shapes: a pyramid is exactly one-third of its corresponding prism. This is a general principle in three-dimensional geometry known as Cavalieri's principle.

Frequently asked questions

What is the formula for the volume of a cone?
The volume of a cone is one-third of the volume of a cylinder with the same base radius and height. The formula is V = (1/3) × π × r² × h, where r is the base radius and h is the perpendicular height. Intuitively, three identical cones can be packed into one cylinder of equal dimensions. This relationship was known to ancient Greek mathematicians and is a standard result in solid geometry.
What is the slant height of a cone and how is it calculated?
The slant height (l) is the distance measured along the surface of the cone from the apex (tip) to the edge of the base circle. It is not the same as the perpendicular height h. By the Pythagorean theorem applied to the right triangle formed by the height, radius, and slant height: l = √(r² + h²). The slant height is needed to calculate the lateral surface area of the cone.
How do you calculate the lateral surface area of a cone?
The lateral surface area (also called the curved surface area) is the area of the cone's sloping surface, excluding the circular base. The formula is A_lateral = π × r × l, where r is the base radius and l is the slant height. If you unrolled the lateral surface of a cone, it would form a sector of a circle with radius l and arc length equal to the circumference of the cone's base (2πr), confirming this formula geometrically.
What is the total surface area of a cone?
The total surface area of a cone includes both the lateral (curved) surface and the circular base. Total SA = π × r × l + π × r², which factors as π × r × (l + r). In many practical applications such as manufacturing or packaging, the total surface area determines the amount of material required to make the cone-shaped object.
What units does the cone volume calculator use?
The calculator is unit-agnostic. Enter radius and height in any consistent unit — centimetres, metres, inches, or feet — and the outputs will be in the corresponding cubed and squared units. If you enter dimensions in centimetres, the volume will be in cubic centimetres (cm³) and the surface areas in square centimetres (cm²). Always use the same unit for both inputs.
What are some real-world examples of cones?
Cones appear frequently in everyday life and engineering. Ice cream cones, traffic cones, and party hats are simple examples. In manufacturing, conical funnels direct material flow. Volcano eruption models, rocket nose cones, and drill bits approximate conical shapes. In architecture, conical roofs appear on turrets and towers. Understanding cone geometry is also important in civil engineering for calculating volumes of stockpiled granular materials like sand and gravel.

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