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.

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 pyramid 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}

Surface area = 2

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

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