How to Calculate the Volume of a Pyramid

To calculate the volume of a pyramid, use the formula $V={\frac {1}{3}}lwh$ , where l and w are the length and width of the base, and h is the height. You can also use the equivalent formula $V={\frac {1}{3}}A_{b}h$ , where $A_{b}$ is the area of the base and h is the height. The method varies 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.

Volume Help

Volume of a Pyramid Cheat Sheet

Method 1

Method 1 of 2:

Pyramid with a Rectangular Base

1
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.^{[1]
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- Remember, $V={\frac {1}{3}}lwh={\frac {1}{3}}A_{b}h$ , so you need to know $l$ and $w$ first.
- $l=4\,{\text{cm}}$
- $w=3\,{\text{cm}}$
2
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.^{[2]
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- Remember, $V={\frac {1}{3}}A_{b}h$ , so you need to know $A_{b}$ . You can find this by plugging in $l=4\,{\text{cm}}$ and $w=3\,{\text{cm}}$ from the previous step.
- $A_{b}=lw$
- $A_{b}=(4\,{\text{cm}})(3\,{\text{cm}})=12\,{\text{cm}}^{2}$
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3
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.
- Remember, $V={\frac {1}{3}}A_{b}h$ , so you need to know $A_{b}h$ . You can find this using $A_{b}$ from the previous step.
- $A_{b}=12\,{\text{cm}}^{2}$
- $h=4\,{\text{cm}}$
- $A_{b}h=(12\,{\text{cm}}^{2})(4\,{\text{cm}})=48\,{\text{cm}}^{3}$
4
Multiply your result so far by ${\frac {1}{3}}$ . Or, in other words, divide by 3. Remember to state your answer in cubic units whenever you're working with three-dimensional space.^{[4]
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- Remember, $V={\frac {1}{3}}lwh={\frac {1}{3}}A_{b}h$ . You can plug in $A_{b}h=48\,{\text{cm}}^{3}$ from the previous step.
- $V={\frac {1}{3}}A_{b}h$
- $V=({\frac {1}{3}})(48\,{\text{cm}}^{3})=16\,{\text{cm}}^{3}$

Method 2

Method 2 of 2:

Pyramid with a Triangular Base

1
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 base is 2 cm and the length of the triangle is 4 cm.^{[5]
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- 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.
- Remember, $V={\frac {1}{3}}lwh={\frac {1}{3}}A_{b}h$ , so you need to know $l$ and $w$ first.
- $l={\text{width of pyramid base}}={\text{base of triangle, or}}\,b=2\,{\text{cm}}$
- $w={\text{length of pyramid base}}={\text{height of triangle, or}}\,h=4\,{\text{cm}}$
2
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: $A_{b}={\frac {1}{2}}bh$ $A_{{b}}={\frac {1}{2}}bh$ .^{[6]
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- Remember, $V={\frac {1}{3}}lwh={\frac {1}{3}}A_{b}h$ , so you need to know $A_{b}$ . You can find this using $b$ and $h$ from the previous step.
- $A_{b}={\frac {1}{2}}bh$
- $A_{b}=({\frac {1}{2}})(2\,{\text{cm}})(4\,{\text{cm}})$
- $A_{b}=({\frac {1}{2}})(8\,{\text{cm}}^{2})$
- $A_{b}=4\,{\text{cm}}^{2}$
3
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.
- Remember, $V={\frac {1}{3}}A_{b}h$ , so you need to know $A_{b}h$ . You can find this using $A_{b}$ from the previous step.
- $A_{b}={\text{area of triangular base}}=4\,{\text{cm}}^{2}$
- $h={\text{height of pyramid}}=5\,{\text{cm}}$
- $A_{b}h=(4\,{\text{cm}}^{2})(5\,{\text{cm}})=20\,{\text{cm}}^{3}$
4
Multiply your result so far by ${\frac {1}{3}}$ . Or, in other words, divide by 3. Your result will show that 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.^{[7]
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- Remember, $V={\frac {1}{3}}lwh={\frac {1}{3}}A_{b}h$ . You can plug in $A_{b}h=20\,{\text{cm}}^{3}$ from the previous step.
- $V=({\frac {1}{3}})A_{b}h$
- $V=({\frac {1}{3}})(20\,{\text{cm}}^{3})=6.67\,{\text{cm}}^{3}$

Community Q&A

Question

How do I calculate the volume of a three square tier pyramid?

Community Answer

Determine the area of the base. If it is a rectangle, that's length x width, if it is a triangle it's 1/2 x the base (one side) x the height (a line perpendicular to the base to the opposite vertex). Determine the height of the pyramid. It is a line perpendicular (straight up) from the base of the pyramid to the opposite vertex. Muliply (1) x (2) and divide by 3. The formula is 1/3 x the area of the base x the height of the pyramid
Question

How do I find the height if given the volume and the base length for a square pyramid?

Donagan

Top Answerer

Triple the volume and divide that by the area of the base (which is the square of the length of an edge).
Question

How do I double the volume of a square pyramid?

Donagan

Top Answerer

Either double the height of the pyramid or double the area of the base. Accomplish the latter by multiplying the length of each side of the base by the square root of 2 (1.414).

Warnings

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.

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