Skill: 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 $\boxed{96\text{ cm}^2}$.