How to Calculate the Surface Area of a Square Pyramid

A pyramid is defined geometrically as a three-dimensional solid with a polygon base where the faces converge on a point. Pyramids may be classified according to the shape of their base. For example, a square pyramid has a square as its base. The following steps will show how to calculate the surface area of a square pyramid by knowing the length of its height and base.

Difficulty: Moderate

Instructions

1. Observe from the definition of a pyramid that the faces are always triangles. Assume for simplicity that the apex of the pyramid is directly above the center of the base, giving the pyramid four sides of equal size.

2. Call a side of the base B, the height H and the surface area of one of its faces S. The surface area of a pyramid is defined as the total area of all of its surfaces including its base and we will call this value A. Our task is now to calculate A in terms of B and H.

3. Determine the surface area of the pyramid in terms of the area of the individual surfaces. Because the base is a square, we have its surface area as B^2. A square pyramid has four sides, so we now have A = B^2 + 4S.

4. Calculate S in terms of B and H. The area of a triangle is given by one half the product of its base and altitude. In this case, the base is simply B so all we need is the value of the altitude L of the pyramid's face in terms of B and H. Note that this line segment is the hypotenuse of a right triangle with H and ½ B.

5. Use the Pythagorean theorem to calculate the altitude of the pyramid's face. We have H^2 + (B/2)^2 = L^2. Therefore, L is the square root of (H^2 + B^2/4.) Since S = ½ B x L, we now have S = ½ B x square root of (H^2 + B^2/4). Using A=B^2 + 4S from step 3, we can replace S with terms of B and H. This gives our final equation of A = B^2 + 2 x B x square root of (H^2 + B^2/4).

Resources

• See a diagram for a right pyramid.

[Saimon]

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