How to Calculate the Volume of a Pyramid

Measure the length and width of the base. In this example, the base length is 4 cm and the width is 3 cm. For a square pyramid, the length and width are the same. Write these values down on paper.

Multiply length and width to calculate the base area. To find the base area of the pyramid, simply multiply 3 cm by 4 cm, which gives 12 cm².

Multiply the base area by the height. The base area calculated above is 12 cm², and the height is 4 cm. Multiply 12 cm² by 4 cm to get 48 cm³.

Divide the result by 3. Alternatively, you can multiply the result by 1/3—both methods will give you the same outcome. 48 cm³ ÷ 3 = 16 cm³. The volume of the pyramid with a height of 4 cm and a rectangular base with side lengths of 4 cm and 3 cm is 16 cm³. Always remember to include the volume unit when working with three-dimensional space.

Determine the length and width of the base. In this case, the length and width must be perpendicular to each other, or you can treat them as the base and height of the triangular base. For this example, the width of the triangle is 2 cm and the length is 4 cm. Write down these values.

• If the length and width are not perpendicular and the height of the triangle is unknown, you may use other methods to calculate the area of the triangle.

Calculate the base area. To determine the base area, substitute the measured values of the base and height of the triangle into the following formula: A = 1/2(b)(h). The detailed calculation is as follows:

• A = 1/2(b)(h)
• A = 1/2(2)(4)
• A = 1/2(8)
• A = 4 cm²

Multiply the base area by the height of the pyramid. The base area calculated above is 4 cm², and the height is 5 cm. Multiply 4 cm² by 5 cm to get 20 cm³.

Divide the result by 3. 20 cm³ ÷ 3 = 6.67 cm³. Therefore, the volume of the pyramid with a height of 5 cm and a triangular base with perpendicular side lengths of 2 cm and 4 cm will be 6.67 cm³.

• In a right pyramid, the height, slant height, and base edge follow the Pythagorean theorem as follows: (base edge ÷ 2)² + (height)² = (slant height)²
• This method can be generalized for objects like pentagonal pyramids, hexagonal pyramids, etc. The general process is as follows: A) Calculate the area of the base; B) Measure the height from the apex of the pyramid to the center of the base; C) Multiply A by B; D) Divide the result by 3.
• In a “regular” pyramid, the slant height, base edge, and lateral edge follow the Pythagorean theorem as follows: (base edge ÷ 2)² + (slant height)² = (lateral edge)²

• A pyramid has three types of height: the slant height which is directed from the apex of the pyramid and perpendicular to the base edge; the actual height, which is the perpendicular distance from the apex to the center of the base, i.e., perpendicular to the base surface; and the lateral edge, which is one side of the lateral triangle, connecting the apex to the base. To calculate the volume, you MUST use the “actual” height.