# Cone Calculator

Calculate cone volume, surface area, slant height, and lateral area from radius and height. Free online cone calculator with all formulas.

## What this calculates

Calculate the volume, surface area, slant height, and lateral area of any cone. Enter the base radius and height to get complete results instantly.

## Inputs

- **Base Radius** — The radius of the circular base.
- **Height** — The perpendicular height from base to apex.

## Outputs

- **Volume** — The volume of the cone (1/3 x pi x r^2 x h).
- **Slant Height** — The distance from base edge to apex along the surface.
- **Lateral Surface Area** — The area of the sloped surface (pi x r x l).
- **Total Surface Area** — Lateral area plus base area.
- **Base Area** — The area of the circular base (pi x r^2).

## Details

A cone is a three-dimensional shape that tapers smoothly from a flat circular base to a point called the apex. It can be thought of as a pyramid with an infinite number of sides.

Cone Formulas

- Volume: V = (1/3) x pi x r^2 x h (one-third of a cylinder's volume)

- Slant Height: l = sqrt(r^2 + h^2) (using the Pythagorean theorem)

- Lateral Surface Area: LSA = pi x r x l (the sloped surface)

- Total Surface Area: TSA = pi x r x l + pi x r^2 (side plus base)

Key Insight

A cone's volume is exactly one-third of the volume of a cylinder with the same base and height. This relationship, proved by Archimedes, connects cones to cylinders and spheres in beautiful ways.

Cones appear in ice cream cones, traffic cones, funnels, volcanoes, and conical roofs. In mathematics, conic sections (ellipses, parabolas, hyperbolas) are curves formed by slicing a cone with a plane.

## Frequently Asked Questions

**Q: How do I calculate the volume of a cone?**

A: Use V = (1/3) x pi x r^2 x h, where r is the base radius and h is the perpendicular height. For example, a cone with radius 3 cm and height 7 cm has volume = (1/3) x pi x 9 x 7 = 65.97 cubic cm. The volume is exactly one-third of a cylinder with the same dimensions.

**Q: What is slant height?**

A: Slant height is the distance from the edge of the base to the apex, measured along the surface of the cone. It forms the hypotenuse of a right triangle with the radius and height as the other two sides. Calculate it using l = sqrt(r^2 + h^2). Slant height is needed for surface area calculations.

**Q: Why is a cone's volume one-third of a cylinder?**

A: This result, first proved rigorously by Archimedes, comes from calculus (or Cavalieri's principle). If you imagine filling a cone with water and pouring it into a cylinder of the same base and height, you would need exactly three cones to fill the cylinder. This 1/3 factor applies to all pyramidal shapes relative to their prismatic counterparts.

**Q: What are conic sections?**

A: Conic sections are curves obtained by intersecting a cone with a plane at different angles. A horizontal cut gives a circle, a tilted cut gives an ellipse, a cut parallel to the side gives a parabola, and a steeper cut gives a hyperbola. These curves are fundamental in physics, astronomy (planetary orbits), and engineering.

**Q: How do I find the height of a cone from its volume and radius?**

A: Rearrange the volume formula: h = 3V / (pi x r^2). For a cone with volume 100 cubic cm and radius 4 cm: h = 3 x 100 / (pi x 16) = 300 / 50.27 = 5.968 cm.

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Source: https://vastcalc.com/calculators/math/cone
Category: Math
Last updated: 2026-04-21