There are formulas that can be used to find both the surface area and the volume of a pyramid. The surface area of a pyramid is the total area of all the surfaces that the pyramid has. To that end, the formula for finding the surface area when all of the side faces are the same is:
SA = (base area) + (1/2) * (perimeter) * (slant height)
The base area is the area of the base and can be determined based on what figure the base is. For example, if the base of the pyramid is a square, the formula for finding the area is:
A = s^2
The perimeter is the distance around the base of the pyramid. The slant height is the diagonal height from the center of one of the base edges to the apex.
If the pyramid has side faces that differ from each other (like in the case of an irregular pyramid), then the surface area equation is:
SA = (base height) + (lateral area)
In this case, you must take each side of the pyramid separately (including the base), find the areas, and then just add them together.
The volume of a pyramid can be found using this formula:
V = (1/3) * (base area) * (height)
The base area is, again, just the area of the base of the pyramid. However, in this case, the height is the length of a line from the apex that makes a right angle with the base.
Example Problems
Let's take a look at some examples.
Example 1
A square pyramid has a height of 9 meters. If the side of the base measures 4 meters, what is the volume of the pyramid?
Since the base is a square, area of the base = 4 * 4 = 16 m^2.
Volume of the pyramid = (1/3) * (base area) * (height)
V = (1/3) * (16) * (9) = 48 m^3
Example 2
What is the surface area of the pyramid discussed in the first example?
In order to find the surface area, we must first find the slant height of the pyramid. Since we know the height and the base length, we can use the Pythagorean Theorem to find the slant length.
You can see that the blue and red lines create a right triangle. The length of the long leg of the triangle is 9 meters, or the height of the triangle. The length of the short leg of the triangle is 2 meters because it is half the length of the base of the triangle (4 m). The red line is the hypotenuse of the triangle and can be calculated using the formula:
a^2 + b^2 = c^2
2^2 + 9^2 = c^2
81 + 4 = c^2
85 = c^2
c = 9.2 m
Now that you know the slant height, you can solve for the surface area of this triangle.
SA = (base area) + (1/2) * (perimeter) * (slant height)
SA = 16 + (1/2) * (16) * (9.2) = 89.6 m^2
Lesson Summary
In geometry, a pyramid is a threedimensional shape that can have any polygon as its base. The corners of the polygon all connect at the apex, or point, of the pyramid. There are formulas that can be used to determine the surface area and the volume of any pyramid. The formula for finding the surface area when all of the side faces are the same is:
SA = (base area) + (1/2) * (perimeter) * (slant height)
The base area is the area of the base and can be determined based on what figure the base is. For example, if the base of the pyramid is a square, the formula for finding the area is:
A = s^2
The perimeter is the distance around the base of the pyramid. The slant height is the diagonal height from the center of one of the base edges to the apex.
If the pyramid has side faces that differ from each other (like in the case of an irregular pyramid), then the surface area equation is:
SA = (base height) + (lateral area)
The volume of a pyramid can be found using this formula:
V = (1/3) * (base area) * (height)
Key Terms
 pyramid: a threedimensional shape that can have any polygon as its base
 triangular pyramid: a pyramid with a triangle as its base
 square pyramid: a pyramid with a square as its base
 pentagonal pyramid: a pyramid with a pentagon as its base
 right pyramid: a pyramid in which the apex of the pyramid is directly above the center of its base
 oblique pyramid: a pyramid in which the apex of the pyramid is not directly above the center of its base
 regular pyramid: a pyramid in which the base of the pyramid is a regular polygon
 irregular pyramid: a pyramid in which the base of the pyramid is an irregular polygon
 surface area: the total area of all the surfaces
 base area: the area of the base
 perimeter: the distance around the base of the pyramid
 slant height: the diagonal height from the center of one of the base edges to the apex
The red arrow is the slant height, or the diagonal height from the center of one of the base edges to the apex.

Learning Outcomes
Upon reviewing this lesson, you should be able to:
 Identify different types of pyramids
 Calculate the volume, area, perimeter, and slant height using pyramid formulas