# How to Find the Lateral Area of a Square Pyramid

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Math problems involving three-dimensional shapes may require us to find the lateral surface area of a square pyramid. The lateral surface area is the sum of the areas of its lateral faces (sides), while the total surface area is the sum of its lateral faces and its base. So in a square pyramid, the lateral faces are the four triangles that form the top and side portions of the shape.

#### TL;DR (Too Long; Didn't Read)

In all of the following calculations we assume that these are right pyramids. This means that the vertex of the pyramid is directly above the center of the base. This means all of the lateral sides are isosceles triangles (or equilateral).

## Solving for Lateral Area

The general formula for the lateral surface area of a regular pyramid is:

\text{lateral area} = 2 \times \text{perimeter of base} \times \text{slant height of pyramid​}

This formula essentially finds the area that makes up each lateral face by the classic formula for area of a triangle, which is:

\text{Area} = \text{base} \times \text{height}

The perimeter acts as the cumulative base for all of the sides, and the slant height of the pyramid is the height of the triangles. We then divide by two to account for the triangle area, and we get the surface area formula.

## Finding the Lateral Area

Calculate the perimeter of the base by multiplying the side length of the pyramid by four – because a square has four equal sides. Finding the slant height of one of the side faces can prove to be more difficult, but in the real world it can often be measured with fairly simple tools.

If the side of a square pyramid – a rectangular pyramid with a square base – measures 6 inches, the perimeter is:

4 \times 6 = 24 \text{\ inches}

The lateral slant height of a square pyramid is the distance from the top of the pyramid to the edge of the base that bisects one of the triangle faces. If the lateral slant height is 8 inches, we can work out

24 \times 8 = 192

#### Tips

• If we are given the height of a square pyramid instead, we can use the pythagorean theorem to find the slant height. Just be careful to use the base of a right triangle within the measurements of the square pyramid.

To find the lateral surface area, we then calculate

We now know that the lateral surface area of a square pyramid with a base perimeter of 24 inches and a lateral slant height of 8 inches is 96 square inches.

#### Tips

• If we already know the area of each of the four lateral faces of a square pyramid, we can work out the lateral surface area by finding the sum of areas of the lateral faces. For example, if the areas of the lateral faces are 10 inches, 10 inches, 7 inches and 7 inches, work out 10 + 10 + 7 + 7 = 34. The lateral surface area is 34 square inches.

## Total Surface Area of a Square Pyramid

After calculating the lateral area of a square pyramid, we can find the total surface area by adding the area of the base to this number. Since the base is a square, we can take one side length of the base (s) and square it to find area.

A_\text{base} = s^2

we then just add this value to the lateral area to find the surface area.

Using the values from the previous example, the base edge has a length of 6 inches, so:

A_\text{base} = 6^2 = 36 \text{\ in}^2

We then add this to the lateral area, 96 square inches to find the surface area of the square pyramid.

\text{Total Surface Area} = 36 + 96 = 132 \text{in}^2

## Lateral Area of a Rectangular Pyramid

A rectangular pyramid introduces another level of complexity when finding the lateral area, but we can still use the same principles as a square-based pyramid. Since the rectangular pyramid still has triangular sides we can use the formula for the area of a triangle, but we need to account for the sides of the rectangle being different. This means that only the triangles across from each other will be equal instead of all four lateral faces being congruent.

To account for this we can calculate the lateral area in pairs of opposite faces so we break up the perimeter into the length (l) and width (w) and their lateral faces.

A_\text{lateral} = \frac{2l \times h_{length} + 2w \times h_{width}}{2}

## Applying These Relationships to Other Pyramids

When the base of the pyramid is a polygon other than a square, this formula for lateral area can still be applied. Whether it is a hexagonal pyramid (with 6 lateral sides) or a pentagonal pyramid (with 5 lateral sides), the general formula at the beginning can be used. Since all pyramids have triangular faces, we can calculate the perimeter by summing the sides of the base, and then we multiply perimeter by the slant height and divide by two to get lateral area.

We can also then calculate the area of the base, and add that, to find the total surface area of the shape.

#### Tips

• The surface area of a triangular pyramid can be found in the same fashion, and if the faces of the triangular pyramid are equilateral, then the area of just one face can be multiplied by 4 to get the total surface area.

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