how to find slant height of a pyramid

Surface Area of a Pyramid

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

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 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.

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

$p = 3 (8) = 24 inches$

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

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.

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.