Section 14.2: Surface Areas of Pyramids

Property

A pyramid is a 3D figure formed by connecting all points on a polygonal base to a single point not on the plane of the base, called the apex or vertex. The polygonal base can be any polygon (triangle, square, pentagon, etc.), and the triangular faces that connect the base to the apex are called lateral faces.

Examples

Property

A triangle is exactly half of a parallelogram with the same base and height. To find its area, choose any side of the triangle as its base (length $b$), and let $h$ be the perpendicular distance from the base to its opposing vertex.
The formula is:

Examples

• A right triangle has legs (base and height) of 5 m and 8 m. Its area is $\frac{1}{2} \times 5 \times 8 = 20$ square meters.
• A triangular sign has a base of 40 cm and a height of 25 cm. Its area is $\frac{1}{2} \times 40 \times 25 = 500$ square cm.
• An obtuse triangle has a base of 10 inches and its corresponding height is 6 inches. The area is $\frac{1}{2} \times 10 \times 6 = 30$ square inches.

Explanation

Any triangle is exactly half of a parallelogram! If you clone a triangle, flip it, and join it to the original, you create a parallelogram. That's why the triangle's area formula is simply one-half of the base times the height.

Property

The surface area of a pyramid is given by the formula:

where $A_{base}$ is the area of the base and $A_{lateral}$ is the total area of all lateral faces.

Examples