Surface Area of a Square Pyramid Calculator
Enter the base edge length and either the vertical height or the slant height of your square pyramid to get the total surface area, lateral surface area, base area, and individual face area. The calculator also derives whichever height dimension you did not enter, shows you the full step-by-step working, and presents a visual breakdown of how the five surfaces fit together. Switch between metric and imperial units at any time.
Formula
Worked example
A pyramid with base edge a = 6 cm and vertical height h = 4 cm: slant height s = sqrt(4^2 + 3^2) = sqrt(25) = 5 cm. Base area = 6^2 = 36 cm^2. Each triangular face = (1/2)(6)(5) = 15 cm^2. Lateral SA = 4 x 15 = 60 cm^2. Total SA = 36 + 60 = 96 cm^2.
What is the surface area of a square pyramid?
A square pyramid has five faces: one square base and four identical isosceles triangles that meet at a single point called the apex. The total surface area is the sum of the base area and the four triangular face areas. Knowing the surface area is useful whenever you need to calculate how much material is required to build or cover a pyramid shape, whether you are estimating roofing material for a hip roof, calculating the amount of paint needed for a decorative model, or solving a geometry problem.
Key dimensions: base edge, height, and slant height
Three lengths fully describe a square pyramid. The base edge length (a) is the side of the square base. The vertical height (h) is the straight-line distance from the centre of the base to the apex, measured perpendicular to the base. The slant height (s) is the distance from the midpoint of a base edge to the apex, measured along the slanted surface of a triangular face. These three are linked by a right triangle: s = sqrt(h^2 + (a/2)^2). This means you only need two of the three to calculate everything else, which is why this calculator lets you enter either h or s alongside the base edge.
Step-by-step: how the formula works
Starting from base edge a and vertical height h: (1) Compute the slant height s = sqrt(h^2 + (a/2)^2). (2) The base area is B = a^2. (3) Each triangular face is a triangle with base a and height s, so its area is FA = (1/2) x a x s. (4) There are four identical faces, so the lateral surface area is LSA = 4 x FA = 2as. (5) Add the base: Total SA = a^2 + 2as. This can be factored as SA = a(a + 2s), which is a compact form useful for mental arithmetic. The famous Pyramid of Giza has a base edge of about 440 m and an original slant height near 280 m, giving a total surface area of roughly 440,000 m^2 - close to 62 football pitches.
Unit handling: metric vs. imperial
Surface area is always expressed in squared length units: cm^2, m^2, in^2, ft^2, and so on. This calculator works in the units you enter, so if you type centimetres, all outputs are in cm^2. Switching between metric and imperial in the unit selector does not convert your entered values; instead it relabels the unit so you can work directly in whichever system suits you. If you need to convert an area result, multiply by 6.4516 to go from in^2 to cm^2, or divide by 6.4516 to go the other way.
Square pyramid surface area reference
| Base edge a (cm) | Height h (cm) | Slant height s (cm) | Base area (cm²) | Lateral SA (cm²) | Total SA (cm²) |
|---|---|---|---|---|---|
| 3 | 2 | 2.5000 | 9 | 15 | 24 |
| 4 | 3 | 3.6056 | 16 | 28.84 | 44.84 |
| 5 | 4 | 4.6098 | 25 | 46.10 | 71.10 |
| 6 | 4 | 5.0000 | 36 | 60 | 96 |
| 8 | 6 | 7.2111 | 64 | 115.38 | 179.38 |
| 10 | 8 | 9.4340 | 100 | 188.68 | 288.68 |
| 12 | 9 | 10.8167 | 144 | 259.60 | 403.60 |
| 440 | 147 | 280.0000 | 193600 | 246400 | 440000 |
Total surface area for common base-edge and height combinations (metric units, cm and cm²).
Frequently asked questions
What is the formula for the surface area of a square pyramid?
The total surface area is SA = a^2 + 2as, where a is the base edge length and s is the slant height. If you know the vertical height h instead of s, first compute s = sqrt(h^2 + (a/2)^2) and then apply the formula. The base area alone is a^2, and the lateral surface area covering the four triangular sides is 2as.
What is the difference between slant height and vertical height?
The vertical height (h) is the perpendicular distance from the apex straight down to the centre of the base. The slant height (s) is the distance from the midpoint of a base edge up to the apex along the face of the pyramid. They are related by s = sqrt(h^2 + (a/2)^2), where a is the base edge. The slant height is always longer than the vertical height for any valid pyramid.
How do I find the lateral surface area of a square pyramid?
The lateral surface area (LSA) covers the four triangular faces and excludes the base. LSA = 2 x a x s, where a is the base edge length and s is the slant height. Equivalently, LSA = 4 x (1/2 x a x s) because each of the four triangles has area (1/2)as. If you only have the vertical height h, compute s = sqrt(h^2 + (a/2)^2) first.
What is the area of just one triangular face?
Each of the four identical triangular faces has an area of FA = (1/2) x a x s, where a is the base edge and s is the slant height. Because the triangle has base a and height (measured along the face) equal to s, the standard triangle-area formula applies directly.
Can the slant height be shorter than half the base edge?
No. For a valid pyramid, the slant height s must be greater than half the base edge (s > a/2). This is because s is the hypotenuse of a right triangle whose shorter leg is a/2. If s were equal to or less than a/2, the apex would be at or below the base, which is not a valid pyramid shape.
How is the Pyramid of Giza used as an example?
The Great Pyramid of Giza originally had a base edge of about 440 m and a slant height of approximately 280 m. Applying the formula: base area = 440^2 = 193,600 m^2; lateral surface area = 2 x 440 x 280 = 246,400 m^2; total surface area = 440,000 m^2. That is roughly 440,000 square metres, or about 62 standard football pitches.