Calculate the surface area of pyramids

Surface area of a pyramid is the sum of the base area and the areas of its triangular lateral faces; for a regular pyramid, this can be expressed as $SA=B+\tfrac12Pl$ , where $B$ is base area, $P$ base perimeter, and $l$ slant height. The distinction between slant height and perpendicular height is essential, with the latter used to find $l$ when needed through right-triangle relationships; treatment is limited to common right or regular pyramids, not generalized oblique or advanced cases.

Detailed Explanation: Calculate the surface area of pyramids

The surface area of a regular pyramid is the area of its base plus the areas of all its triangular faces.

Use the formula

SA=B+\frac12Pl

where:

$B$ is the area of the base,
$P$ is the perimeter of the base,
$l$ is the slant height.

The slant height goes from the midpoint of a base edge to the apex along a triangular face. It is different from the perpendicular height, which goes straight from the apex to the center of the base.

Example

Find the surface area of a square pyramid with:

base side length $6$ cm,
perpendicular height $4$ cm.

Step 1: Find the base area

The base is a square:

B=6^2=36\text{ cm}^2

Step 2: Find the base perimeter

P=4(6)=24\text{ cm}

Step 3: Find the slant height

The perpendicular height, half of the base side, and slant height form a right triangle.

Half of the base side is

\frac{6}{2}=3\text{ cm}

Using the Pythagorean theorem:

l^2=4^2+3^2

l^2=16+9=25

l=5\text{ cm}

Step 4: Substitute into the surface area formula

SA=B+\frac12Pl

SA=36+\frac12(24)(5)

SA=36+60

\boxed{SA=96\text{ cm}^2}

The surface area of the pyramid is $\boxed{96\text{ cm}^2}$ .

Learn by doing: Calculate the surface area of pyramids

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Surface Area - All - Words to Decimal

Earned ?