What is the volume formula for a pyramid?

Volume equals one-third times the base area times the height. For a rectangular base that is V = 1/3 · length · width · height.

How is the surface area of a rectangular pyramid found?

Add the base area to the four triangular faces. Each pair of faces uses a slant height of √(h² + (half the opposite base edge)²).

Does this assume a right pyramid?

Yes. The surface-area result assumes a right rectangular pyramid with the apex directly above the centre of the base.

What is the slant height of a pyramid?

The slant height is the distance from the apex down the middle of a triangular face to the base edge. For a rectangular pyramid there are two slant heights, one for each pair of opposite faces.

What is lateral surface area?

Lateral surface area is the area of the four triangular side faces only, excluding the base. Adding the base area to it gives the total surface area.

Why is a pyramid one-third the volume of a prism?

A pyramid with the same base and height as a prism fills exactly one-third of that prism's volume — a classic result of solid geometry — which is why the volume formula includes the 1/3 factor.

Does the calculator work for a square pyramid?

Yes. A square pyramid is just a rectangular pyramid with equal length and width, so enter the same value for both and the formulas still apply.

Pyramid Calculator — Gera Tools

Pyramid calculator

Enter the base length, width and height of a right rectangular pyramid to get its volume, base area, lateral surface area and total surface area. It is useful for geometry coursework, construction, packaging and any project involving a four-sided pyramid.

How it works

With base length l, width w and vertical height h, the tool computes:

base area = l × w
volume    = (1/3) × l × w × h
slant_l   = √(h² + (w/2)²)        slant_w = √(h² + (l/2)²)
lateral   = l × slant_l + w × slant_w
total     = lateral + base area

The two slant heights reach the midpoints of opposite base edges, which is why each uses half of the other dimension. The apex is assumed to sit directly above the centre of the base.

Example

A pyramid with l = 6, w = 6, h = 9:

Base area: 6 × 6 = 36
Volume: (1/3) × 36 × 9 = 108
Slant heights: √(9² + 3²) = √90 ≈ 9.487 (both, since l = w)
Lateral: 6 × 9.487 + 6 × 9.487 ≈ 113.84
Total surface area: 113.84 + 36 ≈ 149.84

Quantity	Formula	Value
Base area	l × w	36
Volume	⅓·l·w·h	108
Lateral area	l·slant_l + w·slant_w	113.84
Total area	lateral + base	149.84

Everything updates instantly in your browser.