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.
1. Pyramid Type
2. Pyramid Height
Dimensions
Surface Areas
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 Type | Base Area | Slant Height | Volume | Total 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:
- Slant height: l = sqrt(4^2 + (5/2)^2) = sqrt(16 + 6.25) = sqrt(22.25) = 4.717 units
- Volume: V = (1/3) x 5^2 x 4 = (1/3) x 100 = 33.33 cubic units
- Lateral surface area: 2 x 5 x 4.717 = 47.17 square units
- Total surface area: 25 + 47.17 = 72.17 square 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
- Maths and Stats Calculators: full collection of geometry and maths tools.
- Cube Surface Area and Volume Calculator: volume and surface area for a cube.
- Area of a Triangle Calculator: find triangle area from base and height.
- Area of a Triangle (Heron) Calculator: triangle area from three side lengths.
- Annulus Calculator: area and perimeter of a ring shape.
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.