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Pyramid

A pyramid is a polyhedron with a polygonal base and triangular faces that meet at a point called an apex which is not in the plane of the polygonal base.

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Most of the pyramids that are studied in high school are regular pyramids . These pyramids have the following characteristics:

$1)$ The base is a regular polygon.
$2)$ All lateral edges are congruent.
$3)$ All lateral faces are congruent isosceles triangles.
$4)$ The altitude meets the base at its center.

The altitude of a lateral face of a regular pyramid is the slant height. In a non-regular pyramid, slant height is not defined.

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Lateral Surface Area

The lateral surface area of a regular pyramid is the sum of the areas of its lateral faces.

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The general formula for the lateral surface area of a regular pyramid is $L . S . A . = \frac{1}{2} p l$ where $p$ represents the perimeter of the base and $l$ the slant height.

Example 1:

Find the lateral surface area of a regular pyramid with a triangular base if each edge of the base measures $8$ inches and the slant height is $5$ inches.

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The perimeter of the base is the sum of the sides.

$p = 3 (8) = 24$ inches

$L . S . A . = \frac{1}{2} (24) (5) = 60 {inches}^{2}$

Total Surface Area

The total surface area of a regular pyramid is the sum of the areas of its lateral faces and its base. The general formula for the total surface area of a regular pyramid is $T . S . A . = \frac{1}{2} p l + B$ where $p$ represents the perimeter of the base, $l$ the slant height and $B$ the area of the base.

Example 2:

Find the total surface area of a regular pyramid with a square base if each edge of the base measures $16$ inches, the slant height of a side is $17$ inches and the altitude is $15$ inches.

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The perimeter of the base is $4 s$ since it is a square.

$p = 4 (16) = 64$ inches

The area of the base is $s^{2}$ .

$B = 16^{2} = 256 {inches}^{2}$

$\begin{array}{l} T . S . A . = \frac{1}{2} (64) (17) + 256 \\ = 544 + 256 \\ = 800 {inches}^{2} \end{array}$

There is no formula for a surface area of a non-regular pyramid since slant height is not defined. To find the area, find the area of each face and the area of the base and add them.

Volume

The volume of a pyramid equals one-third the area of the base times the altitude (height) of the pyramid. $(V = \frac{1}{3} B h)$ .

Example 3:

Find the volume of a regular square pyramid with base sides $10$ cm and altitude $12$ cm.

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$\begin{array}{l} V = \frac{1}{3} B h \\ V = \frac{1}{3} {(10)}^{2} (12) = 400 {cm}^{2} \end{array}$