# Volume of a Pyramid Calculator

Calculate the volume of a pyramid with square, rectangular, or triangular base. Find volume, base area, and slant height instantly. Free online calculator.

## What this calculates

Find the volume of any pyramid by selecting the base shape and entering the dimensions. This calculator supports square, rectangular, and triangular bases, and also computes the base area and slant height.

## Inputs

- **Base Shape** — options: Square Base, Rectangular Base, Triangular Base — Select the shape of the pyramid's base.
- **Base Side (s)** — min 0 — Side length for square base, or base length for triangular base.
- **Base Width (w)** — min 0 — Width for rectangular base, or height of the triangular base.
- **Pyramid Height (h)** — min 0 — The perpendicular height from base to apex.

## Outputs

- **Volume** — The volume of the pyramid: (1/3) × base area × height.
- **Base Area** — The area of the pyramid's base.
- **Slant Height** — The distance from the apex to the midpoint of a base edge (for square/rectangular base).

## Details

A pyramid is a 3D solid with a polygonal base and triangular faces that meet at a single apex. The volume formula for any pyramid is V = (1/3) x base area x height, where height is the perpendicular distance from the base to the apex.

For a square-based pyramid with side s and height h, the volume is V = (1/3) x s² x h. For a rectangular base with length l and width w, V = (1/3) x l x w x h. For a triangular base with base b and base height h_b, the base area is (1/2) x b x h_b, giving V = (1/6) x b x h_b x h.

The factor of 1/3 in the pyramid volume formula comes from the fact that a pyramid is exactly one-third of a prism with the same base and height. This can be proven by dividing a cube into three congruent pyramids. The slant height is the distance from the apex to the midpoint of a base edge, useful for calculating the lateral surface area.

## Frequently Asked Questions

**Q: What is the formula for the volume of a pyramid?**

A: The volume of any pyramid is V = (1/3) × base area × height. The base area depends on the shape of the base: s² for a square, l × w for a rectangle, or (1/2) × b × h for a triangle.

**Q: Why is the volume one-third of a prism?**

A: A pyramid is exactly one-third the volume of a prism with the same base and height. This can be demonstrated by cutting a cube along its diagonals into three identical square-based pyramids, each with volume equal to one-third of the cube.

**Q: What is slant height?**

A: Slant height is the distance from the apex of the pyramid to the midpoint of a base edge, measured along a lateral face. It differs from the pyramid height, which is the perpendicular distance from the apex to the base. Slant height is used to calculate lateral surface area.

**Q: How is a pyramid different from a cone?**

A: A pyramid has a polygonal base (triangle, square, rectangle, etc.) with flat triangular faces, while a cone has a circular base with a single curved surface. Both share the same volume formula: V = (1/3) × base area × height.

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