Surface Area of a Pyramid Formula

Geometry Formulas: Area, Volume, Perimeter
Formula:

What is the formula for the surface area of a pyramid?

Answer

Surface area of a pyramid
S = a² + 2 · a · l

Here a is the side (or base), l is the slant height of a face, or the length along the solid. The result comes out in square units.

Where the formula comes from

S = a² + 2 · a · l

A square base of a², plus four identical triangular faces. Each face has base a and height l — the slant height of the face — so each is ½·a·l, and four of them give 2·a·l. «Identical» is the load-bearing word: it needs a RIGHT pyramid, whose apex stands over the centre of the square. Lean the apex to one side and the four faces get four different heights, and they have to be added one by one.

Worked examples

Let a = 6, l = 5. Then the surface area of a pyramid = 96 square units.

Let a = 4, l = 3.5. Then the surface area of a pyramid = 44 square units.

The common mistake

Using the pyramid’s own height instead of the slant height of a face. They are different lines: l² = h² + (a/2)². This formula is for a pyramid on a square base; other bases need their own count of faces.

Other quantities: Pyramid

Related Calculations

See Also

Geometry Formulas: Area, Volume, Perimeter

These are the formulas for the area, perimeter, circumference, volume and surface area of the standard shapes — 31 of them. Each page shows not only the formula but where it comes from, what every letter stands for, two worked examples and the mistake that this particular shape invites. For example, it can help you find out what is the formula for the surface area of a pyramid? (The answer is: S = a² + 2 · a · l). Pick the pair — "Area of a circle", for instance — and hit the "Find out" button.
Geometry Formulas: Area, Volume, Perimeter
Formula:

Values from the Formula (l = 5)

If a =Surface area
111 square units
224 square units
456 square units
696 square units
8144 square units
9171 square units

FAQ

What is the formula for the surface area of a pyramid?

Surface area of a pyramid S = a² + 2 · a · l

Where does this formula come from?

A square base of a², plus four identical triangular faces. Each face has base a and height l — the slant height of the face — so each is ½·a·l, and four of them give 2·a·l. «Identical» is the load-bearing word: it needs a RIGHT pyramid, whose apex stands over the centre of the square. Lean the apex to one side and the four faces get four different heights, and they have to be added one by one.

What is the most common mistake here?

Using the pyramid’s own height instead of the slant height of a face. They are different lines: l² = h² + (a/2)². This formula is for a pyramid on a square base; other bases need their own count of faces.

How do I use this formula?

Put your own numbers in place of the letters. With a = 4, l = 3.5, for instance, the formula S = a² + 2 · a · l gives = 44 square units.

See Also