What is the formula for the volume of a pyramid?

V = ⅓ × B × h, where B is the area of the base and h is the perpendicular height from the base to the apex. For a square pyramid with side a this is V = a²h/3; for a rectangular base l × w it is V = lwh/3.

Why is there a 1/3 in the pyramid volume formula?

A pyramid holds one third of the prism with the same base and height. A cube of side a can be cut into three identical square pyramids that each have a face of the cube as the base and one corner as the apex, so each has volume a³/3. Cavalieri's principle extends this to every base shape.

How do I calculate the volume of a 4-sided pyramid?

Multiply the length and width of the base, multiply by the height and divide by 3. A pyramid with a 6 × 4 m base and a 5 m height has V = 6 × 4 × 5 ÷ 3 = 40 m³.

How do I find the volume of a triangular pyramid?

First find the triangle's area, ½ × base × triangle height, then use V = ⅓ × area × pyramid height. With a triangle of base 6 and height 4 (area 12) and a pyramid height of 10, V = 12 × 10 ÷ 3 = 40.

Can I use the slant height to find the volume?

Not directly: the formula needs the perpendicular height. For a square pyramid convert first with h = √(s² − (a/2)²). A pyramid with side 6 and slant height 5 has h = √(25 − 9) = 4 and V = 48.

What is the volume of a truncated pyramid (frustum)?

V = h/3 × (A₁ + A₂ + √(A₁·A₂)), where A₁ and A₂ are the bottom and top areas and h is the distance between them. A square frustum with sides 6 and 3 and height 4 has V = 4/3 × (36 + 9 + 18) = 84.

How do I find the height of a pyramid from its volume?

Rearrange the formula: h = 3V ÷ B. A square pyramid that must hold 48 cm³ on a 6 cm base (B = 36) needs h = 144 ÷ 36 = 4 cm. Set Solve for to Height to do this in the calculator.

Volume of a Pyramid Calculator

Pick the base shape, enter the base size and height, and get the pyramid's volume with V = ⅓ × B × h, plus slant height and surface area, frustum volume, or the height needed for a given volume.

The volume of a pyramid is V = ⅓ × base area × height, so a square pyramid with 6 cm sides and a 4 cm height holds 36 × 4 ÷ 3 = 48 cm³.

Pyramid Dimensions

Solid

Unit

Base Shape

Base side (a)

Solve for

Perpendicular height (h)

Formula Reference

VolumeV = ⅓ · B · h

Square baseB = a²

Rectangle baseB = l × w

Triangle baseB = ½ · b · t

Regular n-gonB = ¼ · n · a² · cot(π/n)

Slant height (square)s = √(h² + (a/2)²)

FrustumV = h/3 · (A₁ + A₂ + √(A₁A₂))

Select a base shape and enter the dimensions to see the volume.

Pyramid Diagram

Pyramid volume formulas by base shape

Right pyramids: the apex sits above the centre of the base. a = base side, h = height, r = apothem (centre to middle of a side).

Base	Base area B	Volume	Slant height
Square (side a)	a²	a²h/3	√(h² + a²/4)
Rectangle (l × w)	l·w	lwh/3	√(h² + w²/4) on the l faces, √(h² + l²/4) on the w faces
Triangle (base b, height t)	½·b·t	b·t·h/6	depends on where the apex sits
Equilateral triangle (side a)	(√3/4)·a² ≈ 0.4330·a²	(√3/12)·a²h	√(h² + a²/12)
Regular pentagon (side a)	≈ 1.7205·a²	≈ 0.5735·a²h	√(h² + r²), r ≈ 0.6882·a
Regular hexagon (side a)	(3√3/2)·a² ≈ 2.5981·a²	(√3/2)·a²h	√(h² + 3a²/4)
Regular n-gon (side a)	¼·n·a²·cot(π/n)	(n·a²·h)/(12·tan(π/n))	√(h² + r²), r = a/(2·tan(π/n))
Frustum (end areas A₁, A₂)	—	h/3·(A₁ + A₂ + √(A₁A₂))	√(h² + (r₁ − r₂)²)

Quick answer

The volume of any pyramid is one third of its base area times its perpendicular height: V = ⅓·B·h. The base area depends on the shape: a² for a square, l·w for a rectangle, ½·b·t for a triangle and ¼·n·a²·cot(π/n) for a regular n-sided polygon. A pyramid holds exactly one third of a prism with the same base and height. Use the perpendicular height, not the slant height; for a square pyramid the slant height s and height h are linked by s² = h² + (a/2)². A truncated pyramid (frustum) with end areas A₁ and A₂ has V = h/3 · (A₁ + A₂ + √(A₁·A₂)).

Examples

Pyramid	Base area B	Height	Volume
Square, side 6	36	4	48
Rectangle 6 × 4	24	5	40
Triangle, base 6, height 4	12	10	40
Regular pentagon, side 4	27.5276	10	91.7588
Regular hexagon, side 2	10.3923	3	10.3923
Frustum, square 6 → 3	36 and 9	4	84

Choose the base shape above, enter its size and the height, and the volume, slant height and surface area update as you type.

Summary

This calculator finds the volume of a pyramid with the universal formula V = ⅓ × base area × height. Choose a square, rectangular, triangular, regular polygon (pentagon, hexagon or any n sides) or custom base. For right pyramids with a square, rectangular or regular base it also gives the slant height, lateral edge, lateral area and total surface area. Switch to a truncated pyramid (frustum) by adding the top size, or solve backwards for the height or base side that gives a target volume. Results convert to litres, gallons, cubic feet or cubic yards.

How it works

Choose a unit and the base shape: square, rectangle, triangle, regular polygon (enter the number of sides) or a known base area.
Enter the base dimensions and the perpendicular height (the straight distance from the base to the apex).
For a truncated pyramid, pick Frustum and enter the top side, top length and width, or top area.
To work backwards, set Solve for to Height or Base side and enter the volume you need.
Read the volume, base area, slant height, lateral and total surface area, and the volume in litres or gallons.

Use cases

Solving geometry homework on square, rectangular, triangular and pentagonal pyramids.
Finding the volume of a truncated pyramid such as a hopper, planter or footing.
Working out the height a pyramid needs to hold a given volume.
Estimating concrete, sand or fill for pyramid-shaped forms and stockpiles.
Checking surface area for the material needed to cover a pyramid roof.
Comparing a pyramid with the prism of the same base (one third of its volume).

Frequently Asked Questions

Last updated: 2026-10-10 · Reviewed by Nham Vu