Lesson 3: Surface Areas of Pyramids

Property

The surface area ($SA$) of a square pyramid is the sum of the area of its square base and the area of its four triangular lateral faces.

where $s$ is the side length of the square base and $l$ is the slant height of the pyramid.

Examples

• A square pyramid has a base side length of $5$ cm and a slant height of $8$ cm. Its surface area is $SA = 5^2 + 2(5)(8) = 25 + 80 = 105 \text{ cm}^2$.
• A square pyramid has a base with an area of $36 \text{ m}^2$ and a slant height of $10$ m. The side length is $\sqrt{36} = 6$ m. Its surface area is $SA = 36 + 2(6)(10) = 36 + 120 = 156 \text{ m}^2$.

Explanation

To find the surface area of a square pyramid, you must calculate two components: the area of the square base and the total area of the four identical triangular faces. The area of the base is found by squaring its side length ($s^2$). The area of the four lateral faces is found by multiplying the area of one triangle ($\frac{1}{2} \times \text{base} \times \text{slant height}$) by four. The total surface area is the sum of the base area and the lateral area.

Property

The surface area ($S.A.$) of a triangular pyramid is the sum of the area of its triangular base ($A_{base}$) and the areas of its three triangular lateral faces.

Examples

• A triangular pyramid has a base with an area of $15 \text{ cm}^2$ and three lateral faces each with an area of $12 \text{ cm}^2$. The total surface area is $S.A. = 15 + 12 + 12 + 12 = 51 \text{ cm}^2$.
• A regular triangular pyramid has an equilateral triangle base with a side length of $6$ m and an area of $15.6 \text{ m}^2$. Its slant height is $8$ m. The area of each lateral face is $\frac{1}{2}(6)(8) = 24 \text{ m}^2$. The surface area is $S.A. = 15.6 + 3(24) = 15.6 + 72 = 87.6 \text{ m}^2$.

Explanation

To find the surface area of a triangular pyramid, you must calculate the area of four distinct triangles: the base and the three lateral faces. First, find the area of the triangular base. Next, calculate the area of each of the three triangular faces that meet at the apex, often using the slant height. Finally, add all four areas together to get the total surface area of the pyramid.