A **pyramid** is a polyhedron formed from a polygonal base and a set of triangular faces. The triangular faces connect to one side of the base and all join together at the apex.

- Use the sliders to adjust the height and side length of the square-based pyramid
- Then drag the Slide to unravel slider to create the net

- What is the area of the base?
- What is the area of one of the triangular sides?
- What is the total surface area?
- Write a formula for surface area.
- Write another formula using the variables B for base area and p for perimeter of the base.

The surface area of a square pyramid can be found as:

\text{SA of a square pyramid} = \text{Area of square base}+ \text{Area of all triangles}

Because the base is a square all 4 of the triangular faces are congruent.

Rewriting the repeated addition as multiplication shows we have four times the area of one triangle plus the area of the square base: SA=4 \cdot \dfrac{1}{2} l s + ls

Using the commutative property, we can rewrite this as: SA = \dfrac{1}{2} \cdot l \cdot 4s + ls

Notice that 4s is the perimeter of the square base, which we can call p, and ls is the area of the square base, which we can call B.

Putting that all together, we can develop a formula for the surface area of a square-based pyramid:

\displaystyle SA=\dfrac{1}{2}lp+B

\bm{l}

slant height

\bm{s}

side length

\bm{p}

perimeter of base

\bm{B}

area of base

A square-based pyramid has a base side length of 30 meters and a slant height of 25 meters.

a

Create the net of the pyramid.

Worked Solution

b

Use the net to find the surface area.

Worked Solution

A square-based pyramid has a base side length of 9 centimeters and a height of 3 centimeters. Calculate the surface area of the pyramid.

Worked Solution

A square pyramid has a surface area of 96\text{ ft}^{2}. Each triangular face has an area of 15\text{ ft}^2.

Find the side length of the base of the pyramid, correct to the nearest foot.

Worked Solution

Joan visited Pyramid of Giza, which is a square pyramid. At the gift shop, she bought a miniature replica of the pyramid that has a slant height of 6 inches and a base length of 4 inches. What is the surface area of the replica?

Worked Solution

Idea summary

A square pyramid is a polyhedron with a square base and four faces that are congruent triangles with a common vertex.

To find the surface area of a square pyramid, we can use the equation:

\text{Surface area of a square pyramid} = \dfrac{1}{2}lp + B

If we are given the perpendicular height, we can find the slant height by using the Pythagorean theorem: a^2+b^2=c^2.