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Surface area of a pyramid

Definition: The number of square units that will exactly cover the surface of a pyramid.
Try this Drag the orange dots to adjust the base size and height of the pyramid and note how the area changes.

The total surface area of any polyhedron, is sum of the surface areas of each face.
In the case of a right pyramid, the side faces are all the same, so we can simply find the area of one and multiply by the number of faces. Once we add the area of the base, we have the total surface area.

The base

In the figure above, the base is a square. So to find its area we multiply the side length by itself. The base however can be any polygon. To find the area of a polygon see Area of a regular polygon.

In the figure above, click on 'reset'. The base side length is 10, so since the base is a square in this example, the base area is 102 or 100.

The sides

The sides of a pyramid are triangles. There are various ways to find the area of triangles (see Area of triangles.) We find the area of one face, then multiply by the number of faces.

In the figure above press 'reset'. We see from the front face that the base of the triangle is 10. We are also given the height* of the triangle - 11.    Recall that the area of a triangle is half the base times height, so each face has an area of 55. (half of 11 times 10). The total for the four faces is 220. (4 times 55).

*This is also called the "slant height" of the pyramid - to distinguish it from the perpendicular height.

Total Area

So the total surface area of the above pyramid is
 Area of the base 100 Area of the four faces = 4 times 55 220 TOTAL 320

As a formula

Since the base of a pyramid can be any polygon, and you may be given various different measurements, it's best to follow the method above to find the area. But in the particular case of a right square pyramid with the base side and slant height given, the area is given by the formula Where:
b is the side length of the base
h is the slant height.

By combining the 4 and the 2, this simplifies a little to The first one is better because it shows more clearly how it is made up from its parts - the base area plus four face areas.

Oblique and irregular pyramids

If the pyramid is oblique (leaning to one side) or the base is irregular, there is no straightforward way to find the surface area. See Area of irregular polygons to find the base area. Each triangular face will be a different shape and size, so you would have to find the area of each using whatever measurements you are given. See Area of triangle for various methods.

Things to try

• In the figure above, click "hide details".
• Drag the orange dots to set the base size and height of the pyramid.
• Calculate the surface of the pyramid using the formula
• Click "show details" to check your answer.
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