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In these lessons, we will learn

- how to find the surface area of any pyramid.
- how to find the surface area of a regular pyramid.
- how to find the surface area of a square pyramid.

- how to find the surface area of a pentagonal and a hexagonal pyramid.
- how to find the surface area of a pyramid when the slant height is not given.

Related Topics: More Geometry Lessons

A pyramid is a solid with a polygonal **base** and several triangular **lateral faces**. The lateral faces meet at a common **vertex**. The number of lateral faces depends on the number of sides of the base. The **height** of the pyramid is the perpendicular distance from the base to the vertex.

A regular pyranid has a base that is a regular polygon and a vertex that is above the center of the polygon. A pyramid is named after the shape of its base. A rectangular pyramid has a rectangle base. A triangular pyramid has a triangle base.

We can find the surface area of any pyramid by adding up the areas of its lateral faces and its base.

Surface area of any pyramid = area of base + area of each of the lateral faces

If the pyramid is a regular pyramid, we can use the formula for the surface area of a regular pyramid.

whereSurface area of regular pyramid = area of base +

ps

If the pyramid is a square pyramid, we can use the formula for the surface area of a square pyramid.

Surface area of square pyramid =

b^{2}+ 2bs

where *b* is the length of the base and *s* is the slant height.

* Example: *

Calculate the surface area of the following pyramid.

* Solution: *

Sketch a net of the above pyramid to visualize the surfaces.

Since the given pyramid is a square pyramid, we can use any of the above formulas.

Using the formula for the surface area of any pyramid

Area of base = 6 × 6 = 36 cm^{2}

Area of the four triangles = × 6 × 12 × 4 = 144 cm^{2}

Total surface area = 36 + 144 = 180 cm^{2}

Using the formula for a regular pyramid

Surface area of regular pyramid = area of base + *ps*

= 6 × 6 + × 6 × 4 × 12 = 180 cm^{2}

Using the formula for a square pyramid

Surface area of square pyramid = *b*^{2} + 2*bs*

= 6 × 6 + 2 × 6 × 12 = 180 cm^{2}

The following videos show how to find the surface area of a pyramid by adding up the area of each surface.

These videos show how to find the surface area of a square pyramid using the formula

Surface area = 2*bs* + *b*^{2} where *b* is the length of the base and *s* is the slant height.

These videos show how to calculate the surface area of a regular pyramid using the formula

surface area = area of base + 1/2 × perimeter of base × slant height.

This video provides a specific example of how to find the surface area of a pyramid, given base edge and height. The base is a pentagon. It shows how to find the apothem and slant height.

How to find the surface area of a pentagonal pyramid with the known apothem.

How to find the surface area of a hexagonal pyramid with known side length.

This video shows how to find the lateral surface area of a pyramid when the slant height is not given. (Part 1)

This video shows how to find the total surface area of a pyramid when the slant height is not given. (Part 2)

This video shows how to calculate the surface area of a square pyramid when the slant height is not given.

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.