Pyramid Volume and Surface Area Calculator

This calculator works out the volume and surface area of a right pyramid, the shape used in architecture, roof design, civil engineering and school geometry. You start by choosing the base shape, square, rectangular, or a right-triangle base, then enter the base dimensions (a side length, or a length and width, or the triangle's base and height) along with the pyramid's perpendicular height, the vertical distance from the apex down to the base. You can also pick a unit label, such as metres, centimetres or feet, for the results to display in. Once you enter your figures, the calculator instantly returns four results: the volume in cubic units, the total surface area covering all faces including the base, the lateral surface area of just the sloping triangular faces, useful if you are cladding or painting only the sides, and the slant height, the distance from the apex to the midpoint of a base edge along the face. A dimensions and surface-area breakdown, plus a full worked example with every step shown, appear below the results so you can check how each figure was reached. Use it to check homework, plan materials for a pyramid-shaped structure or roof, estimate stockpile volumes, or satisfy curiosity about a monument such as the Great Pyramid of Giza. The formulas assume a right pyramid with the apex directly above the centre of the base; oblique pyramids, where the apex sits off-centre, are not covered here.

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Standard geometry  Formulas based on standard Euclidean geometry for right pyramids with a flat polygonal base and apex directly above the centroid.

1. Pyramid Type

units

2. Pyramid Height

units
Please enter positive numbers for all dimensions.

Results

Volume
33.33
units3
Total Surface Area
72.17
units2
Lateral Surface Area
47.17
units2
Slant Height
4.717
units

Dimensions

Base shapeSquare base (side 5 units)
Base area25 units2
Perpendicular height (h)4 units
Slant height (l)4.717 units
Volume33.3333 units3

Surface Areas

Base area25 units2
Lateral surface area47.1699 units2
Total surface area72.1699 units2

Worked Example

Pyramid Formulas

A right pyramid has a flat polygonal base and an apex directly above the centre of the base. The perpendicular height (h) is the straight-line distance from the apex to the base. The slant height (l) runs from the apex to the midpoint of a base edge along the face of the pyramid.

Base TypeBase AreaSlant HeightVolumeTotal Surface Area
Square (side a) a^2 sqrt(h^2 + (a/2)^2) (1/3) x a^2 x h a^2 + 2al
Rectangle (l x w) l x w Two values: sqrt(h^2+(w/2)^2) and sqrt(h^2+(l/2)^2) (1/3) x l x w x h lw + l x sl_w + w x sl_l
Right triangle (base b, height t) (1/2) x b x t Varies per face (1/3) x (1/2) x b x t x h Base area + sum of three lateral faces

How Volume Is Calculated

All pyramids share the same volume formula: V = (1/3) x Base Area x h. This holds regardless of the base shape. The factor of one third arises because a pyramid fills exactly one third of the volume of a prism with the same base and height. You can verify this by filling a pyramid-shaped container three times to fill an equivalent prism.

How Surface Area Is Calculated

The total surface area of a pyramid equals the base area plus the combined area of all the triangular lateral faces. Each lateral face is a triangle with a base equal to one edge of the pyramid's base and a height equal to the slant height (l). For a square-base pyramid with side a, there are four identical lateral faces, each with area (1/2) x a x l, giving a lateral area of 2al. Total surface area = a^2 + 2al.

For a rectangular-base pyramid, the two pairs of lateral faces have different slant heights (one slant height for the length direction, another for the width direction). The calculator handles both.

Square Pyramid Worked Example

For a square-base pyramid with base side a = 5 units and height h = 4 units:

These results match the calculator's defaults.

Practical Uses

Pyramid calculations appear in architecture (roof design, monuments), civil engineering (earthworks and stockpile volumes), and manufacturing (cone and hopper design). The Great Pyramid of Giza has a square base of approximately 230.4 m per side and an original height of about 146.5 m, giving a volume of roughly 2.58 million cubic metres.

Related Calculators

Sources and method: Standard Euclidean geometry formulas for right pyramids. Weisstein, Eric W. "Pyramid." MathWorld. Coxeter, H.S.M. Introduction to Geometry, 2nd ed. (1969).

This calculator applies to right pyramids where the apex is directly above the centroid of the base. Oblique pyramids (where the apex is offset) use the same volume formula but have different lateral face areas not covered here.

Frequently asked questions

What is the formula for the volume of a pyramid?

The volume of any pyramid is one third of the base area multiplied by the perpendicular height: V = (1/3) x Base Area x h. For a square-base pyramid with side a and height h, this gives V = (1/3) x a^2 x h. For a rectangular base with length l and width w, it is V = (1/3) x l x w x h. For a triangular base, the base area is (1/2) x base x height of the triangle.

How do you find the slant height of a pyramid?

The slant height (l) runs from the apex down to the midpoint of a base edge. For a square-base pyramid with base side a and perpendicular height h, slant height = sqrt(h^2 + (a/2)^2). For a rectangular pyramid, there are two different slant heights: one for the length side, sqrt(h^2 + (w/2)^2), and one for the width side, sqrt(h^2 + (l/2)^2). The slant height is needed to calculate the lateral surface area.

What is the difference between lateral surface area and total surface area of a pyramid?

Lateral surface area is the combined area of the triangular faces only, not including the base. Total surface area adds the base area to the lateral area. For a square-base pyramid: lateral area = 2 x a x slant height, and total surface area = a^2 + 2 x a x slant height. If you are painting the outside of a solid pyramid (base included), use total surface area. If you are calculating just the sloping faces (for example, for a tent or a roof), use lateral surface area.

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