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# Cone calculator - volume, surface area, generatrix, radius of base

The cone calculator will help you calculate the volume of the cone, the total area of the cone, the side area of the cone, the base area of the cone, the generatrix length and the radius of the cone base, and the volume of the sphere described on the cone.

## Slant height (generatrix) of the cone - calculator

$$l = \sqrt {h^2 + r^2}$$

## Lateral surface area of the cone - calculator

$$P_{b} = \pi \cdot r \cdot l$$

## Base area of the cone - calculator

$$P_{p} = \pi \cdot r^2$$

## Surface area of the cone - calculator

$$P_{c} = P_{b} + P_{p}$$ $$P_{c} = \pi r l + \pi r^2 = \pi r(r + l)$$

## Volume of the cone - calculator

$$V = \frac{1}{3}P_{p} \cdot h = \frac{\pi r h}{3}$$

## Volume of the sphere described on the cone - calculator

$$V_{k} = \frac{1}{6} \pi \frac{l^6}{(l^2 - r^2) \sqrt{l^2 - r^2}}$$

Cone - information

Right circular cone is a convex solid formed by the rotation of a right triangle around one of the legs (cathetuses). This cathetus creates the height (h) of the cone, the other leg (cathetus) becomes the radius of the base (r), and the hypotenuse - slant height of the cone (l).

Slant height of the cone

Slant height follows from the Pythagorean theorem:
$$l=\sqrt {h^{2}+r^{2}}$$

Lateral surface area of the cone

Lateral surface of the cone, after it is stretched on the plane, forms a circular segment with a radius R = l as the slant height of the cone and the arc length equal to the circumference of the cone base L=2πr
A circular segment with radius R and arc length L has an area: $$P=\frac {1}{2}LR$$ hence $$P_{b}=\frac {1}{2}LR=\frac {1}{2}2\pi rl=\pi rl$$

Base area of the cone

The base of the cone is formed by a circle with a radius r. $$P_{p} = \pi r^2$$

Surface area of the cone

$$P_{c} = P_{b} + P_{p}$$ hence $$P_{c} = \pi r l + \pi r^2 = \pi r(r + l)$$

Volume of the cone

$$V = \frac{1}{3}P_{p}h = \frac{\pi r^2 h}{3}$$

Volume of the sphere described on the cone

$$V_{k} = \frac{1}{6} \pi \frac{l^6}{(l^2 - r^2) \sqrt{l^2 - r^2}}$$

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