What is the formula for the volume of a pyramid?

Every pyramid — regardless of base shape — follows the same rule: V = (1/3) times base area times height. The base area formula changes with the shape (b-squared for a square, length times width for a rectangle, Heron's formula for a triangle, (3 times root 3 / 2) times s-squared for a regular hexagon), but the one-third factor is universal. This is a consequence of Cavalieri's principle: any pyramid with the same base area and height as another has the same volume.

How is slant height different from perpendicular height?

Perpendicular height (h) is the straight-line distance from the apex straight down to the base, measured at a right angle. Slant height (l) is the distance from the apex to the midpoint of a base edge, running along the face of the pyramid. For a square pyramid, slant height = root(h-squared + (b/2)-squared). Slant height is what you need for calculating lateral surface area; height is what you need for volume.

How is the lateral surface area of a pyramid calculated?

Each triangular face has area = (1/2) times base edge times slant height. For a square pyramid that gives 4 times (1/2) times b times l = 2bl. For a hexagonal pyramid it is (1/2) times perimeter times slant height. Total surface area = base area + lateral area. This calculator shows both values separately so you can use whichever you need.

Can I use this for an oblique pyramid?

This calculator models right pyramids, where the apex sits directly above the centre of the base. For oblique pyramids the volume formula V = (1/3) times base area times height still holds as long as h is the true perpendicular height, but the slant height and lateral area figures will differ because the faces are no longer congruent. Use the volume result with caution and verify surface areas manually for oblique cases.

What is the volume of a pyramid with a square base of 6 m and height 10 m?

Base area = 6 times 6 = 36 m-squared. Volume = (1/3) times 36 times 10 = 120 m-cubed. Slant height = root(10-squared + 3-squared) = root(109) = approximately 10.44 m. Lateral area = 4 times (1/2) times 6 times 10.44 = approximately 125.3 m-squared. Total surface area = 36 + 125.3 = approximately 161.3 m-squared.

How do I find the height of a pyramid if I know the volume and base?

Rearrange the volume formula: h = 3V divided by base area. For example, if a square pyramid with a 5 m base has a volume of 100 m-cubed, then base area = 25 m-squared and h = 3 times 100 divided by 25 = 12 m. Use the "Solve for Height" mode in this calculator to do this instantly.

Pyramid Volume Calculator

A pyramid is one of geometry’s most ancient and enduring forms — from the monuments of Giza to the glass pyramid at the Louvre. Whether you are a student working through a geometry problem set, an architect sizing a feature roof, or an engineer estimating the volume of a stockpile, this calculator gives you every key measurement in one step: volume, base area, lateral surface area, total surface area and slant height.

It handles four base shapes — square, rectangular, triangular and regular hexagonal — and supports reverse calculation (find the height or base side needed to hit a target volume).

The core formula

Every pyramid shares a single volume rule:

V = (1/3) x A_base x h

where A_base is the area of the base polygon and h is the perpendicular height (the straight-line distance from the apex down to the base plane, at a right angle). The one-third factor is not an approximation — it is exact, and it holds for any pyramid regardless of base shape or whether the apex is directly above the centre (right pyramid) or offset (oblique pyramid), as long as h is measured perpendicularly.

The base area formula is the only part that changes with the shape:

Square base: A = b x b
Rectangular base: A = l x w
Triangular base (Heron’s formula): A = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2
Regular hexagonal base: A = (3 x sqrt(3) / 2) x s^2

Slant height and surface area

Slant height (l) is the distance from the apex to the midpoint of a base edge, measured along the face. For a right square pyramid:

l = sqrt(h^2 + (b/2)^2)

Each triangular face then has area (1/2) x base edge x l, so the total lateral area of a square pyramid is 2 x b x l. Total surface area adds the base: A_total = A_base + A_lateral.

The slant height matters for construction: if you are cutting panels for a pyramidal roof or a model, the face height you need to mark out is the slant height, not the perpendicular height.

Worked example — Great Pyramid proportions

The Great Pyramid of Giza originally had a square base of about 230.4 m and a height of 146.5 m.

Base area = 230.4 x 230.4 = 53,084 m-squared (approximately 5.3 hectares)
Volume = (1/3) x 53,084 x 146.5 = approximately 2.6 million m-cubed
Slant height = sqrt(146.5-squared + 115.2-squared) = sqrt(21,462 + 13,271) = sqrt(34,733) = approximately 186.4 m
Lateral area = 4 x (1/2) x 230.4 x 186.4 = approximately 86,000 m-squared

Enter those values into the calculator to verify every step instantly.

Reverse calculation

Need to design a pyramid that holds exactly 500 litres (0.5 m-cubed) with a 1 m square base? Switch “Solve for” to Height:

h = 3V / b^2 = 3 x 0.5 / (1 x 1) = 1.5 m

This is useful in packaging design, architecture, landscape earthworks and storage tank sizing.

Units

The calculator accepts any consistent linear unit — millimetres, centimetres, metres, kilometres, inches, feet or yards. Switch the unit selector and all labels update. Volume results are in the selected unit cubed; area results in the unit squared.

Accuracy note

Results are computed in 64-bit floating-point arithmetic (IEEE 754 double precision) — the same precision used by scientific calculators. For very large or very small values the display switches to scientific notation automatically. The triangular base uses Heron’s formula, which can lose precision when the triangle is very flat (one angle close to 180 degrees); for such degenerate cases use higher-precision tools or reframe the problem.

Every calculation runs locally in your browser. No dimensions are transmitted to any server.