How to Calculate the Volume of a Pyramid

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Last Updated: August 12, 2024 Fact Checked

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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

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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.^{[4]
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- 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.^{[5]
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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.
- $V={\frac {1}{3}}A_{b}h$
- $V=({\frac {1}{3}})(48\,{\text{cm}}^{3})=16\,{\text{cm}}^{3}$
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Joseph Meyer
Math Teacher
Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.

Joseph Meyer
Math Teacher

To find a pyramid's volume, use the formula (1/3) $*$ Base Area $*$ Height. Measure a pyramid's height from its tip to the base's center. Next, find the base area using the correct formula for the base shape, whether a triangle, square, or rectangle. Finally, input these values into the formula to calculate volume.

Method 2

Method 2 of 2:

Pyramid with a Triangular Base

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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.^{[6]
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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: $A_{b}={\frac {1}{2}}bh$ $A_{{b}}={\frac {1}{2}}bh$ .^{[7]
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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.^{[8]
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- 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.^{[9]
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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.
- $V=({\frac {1}{3}})A_{b}h$
- $V=({\frac {1}{3}})(20\,{\text{cm}}^{3})=6.67\,{\text{cm}}^{3}$

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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

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

Donagan

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

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How do I double the volume of a square pyramid?

Donagan

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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).

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Tips

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.

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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

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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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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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