How to Calculate the Volume of a Pyramid

To calculate the volume of a pyramid, all you have to do is find the product of the area of the base and the height and multiply the result by 1/3. The method will vary slightly depending on whether the pyramid has a triangular or a rectangular base. If you want to know how to calculate the volume of a pyramid, just follow these steps.

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Volume of a Pyramid Cheat Sheet

Pyramid with a Rectangular Base

Find the length and width of the base. In this example, the length of the base is 4 cm and the width is 3 cm. If you're working with a square base, the method is the same, except the length and width of the square base will be equal. Write down these measurements.

Multiply the length and width to find the area of the base. To get the area of the base, simply multiply 3 cm by 4 cm. 3 cm x 4 cm = 12 cm²^[1]

Multiply the area of the base by the height. The area of the base is 12 cm² and the height is 4 cm, so you can multiply 12 cm² by 4 cm. 12 cm² x 4 cm = 48 cm³

Divide the result by 3. This is the same as multiplying the result by 1/3. 48 cm³/3 = 16 cm³. The area of a pyramid with a height of 4 cm and a rectangular base with a width of 3 cm and a length of 4 cm is 16 cm³. Remember to state your answer in cubic units whenever you're working with three-dimensional space.

Pyramid with a Triangular Base

Find the length and width of the base. The length and width of the base must be perpendicular to each other for this method to work. They can also be considered the base and height of the triangle. In this example, the width of the triangle is 2 cm and the length of the triangle is 4 cm. Write these measurements down.^[2]
- If the length and width are not perpendicular and you don't know the height of the triangle, there are a few other methods you can try to calculate the area of a triangle.

Calculate the area of the base. To calculate the area of the base, just plug the base and height of the triangle into the following formula: A = 1/2(b)(h). Here's how you do it:
- A = 1/2(b)(h)
- A = 1/2(2)(4)
- A = 1/2(8)
- A = 4 cm²

Multiply the area of the base by the height of the pyramid. The area of the base is 4 cm² and the height is 5 cm. 4 cm² x 5 cm = 20 cm³.

Divide your answer by 3. 20 cm³/3 = 6.67 cm³. Therefore, the volume of a pyramid with a height of 5 cm and a triangular base with a width of 2 cm and a length of 4 cm is 6.67 cm.³

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In a square pyramid, the true height, slant height, and length of the edge of the base face are all related by the Pythagorean theorem: (edge ÷ 2)² + (true height)² = (slant height)²

This method can be further generalized to such objects as pentagonal pyramids, hexagonal pyramids, etc. The overall process is: A) calculate the area of the base shape; B) measure the height from the tip of the pyramid to the center of the base shape; C) multiply A with B; D) divide by 3.

In all regular pyramids, the slant height, edge height, and edge length are also related by the Pythagorean theorem: (edge ÷ 2)² + (slant height)² = (edge height)²

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Pyramids have three kinds of height --- a slant height, down the center of the triangular sides; a true height or perpendicular height, that goes from the tip of the pyramid to the center of the base face; and an edge height, that goes down one edge of the triangular sides. For volume, you MUST use the true height