Volume of a Pyramid

The volume of a pyramid is how many cubic units will fit inside of the pyramid. This is determined using a special formula for most types of pyramids. Students can also use a certain formula to determine the surface area to fully solve the pyramid.

Square Pyramids and Finding the Volume

Square pyramids have a square for the base, meaning there are 4 sides to the pyramid and all 4 sides of the base are equal. To determine the volume, it’s necessary to know the length of the sides and the height of the pyramid. Once these are known, the formula to find the volume is (1/3)bh, where b is the area of the base and h is the height of the pyramid. Solve an example with a base that has 4 for the length of the sides and 7 for the height.

The first step is to find the area of the base. Since the base is square, the area is 4^2 or 16. Putting this into the volume formula for the pyramid, the equation is V=(1/3)(16)(7). In this example, the volume is 37 1/3.

Other Types of Pyramids

Not all pyramids are square pyramids. The base can be any polygon, including rectangles and triangles. The pyramids can also be right or oblique. With a right pyramid, the point where all sides meet is directly above the center point of the base, creating a right angle with the base. With an oblique pyramid, the point where all sides meet at the top is not directly over the center of the base. Additionally, pyramids can be regular or irregular. Regular pyramids have a regular polygon, so all sides and angles are the same, for the base while irregular pyramids have an irregular polygon for the base.

Finding the Volume for Other Pyramids

Finding the volume for other pyramids works the same as it does for the square pyramids, but the methods for finding the base area will vary. Use the correct formula to find the area for the type of base and then solve for the volume. With oblique pyramids, use the height of the pyramid as if it’s a right angle with the base.

Finding the Surface Area of a Pyramid

Finding the surface area of a pyramid depends on whether the sides are all the same or all different. When the sides are all the same, the formula for the surface area is b+(1/2)ps, with b being the base area of the pyramid, p being the perimeter (all sides added together), and s being the slant height. The slant height is the distance from the base to the center point along the side triangle, not the height of the pyramid. With irregular pyramids, the surface area is found by adding the areas of all the sides and the base.

Students often think of the great pyramids of Egypt when they imagine pyramids. These are square pyramids and the above information can help the student determine just how big those pyramids are. When the student knows the volume and surface area, it’s easy to imagine just how large those pyramids are.