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1Pyramid with a Rectangular Base
2Pyramid with a Triangular Base
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Reviewed byGrace Imson, MA
Last Updated: April 12, 2024References
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To calculate the volume of a pyramid, use the formula 
, where l and w are the length and width of the base, and h is the height. You can also use the equivalent formula 
, where 
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.
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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]
- Remember,
, so you need to know and first.
- Remember,
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]2[3]
- Remember, , so you need to know . You can find this by plugging in and from the previous step.
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- Remember,
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, , so you need to know . You can find this using from the previous step.
- Remember,
4
Multiply your result so far by
. Or, in other words, divide by 3. Remember to state your answer in cubic units whenever you're working with three-dimensional space.[4]- Remember, . You can plug in from the previous step.
EXPERT TIP
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 TeacherTo 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.- Remember,
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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.[5]
- 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, , so you need to know and first.
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:
.[6]- Remember, , so you need to know . You can find this using and from the previous step.
- Remember,
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, , so you need to know . You can find this using from the previous step.
- Remember,
4
Multiply your result so far by
. 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]3- Remember, . You can plug in from the previous step.
- Remember,
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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
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).
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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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About This Article
Reviewed by:
Grace Imson, MA
Math Teacher
This article was reviewed by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 754,700 times.
28 votes - 88%
Co-authors: 51
Updated: April 12, 2024
Views:754,700
Categories: Featured Articles | Volume
Article SummaryX
To calculate the volume of a pyramid, you need to know its height and the area of the base. Once you have that information, you can find the volume using the formula V (volume) = 1/3 x Ab (the area of the base) x h (height). If the pyramid has a square or rectangular base, simply multiply the width of the base by its length to find the area. Then, multiply the area of the base by the height of the pyramid, and multiply the result by 1/3—which is the same as dividing by 3. For instance, if your pyramid has a square base that is 3 inches long by 3 inches wide, and a height of 4 inches, the volume would be (3 x 3 x 4)/3, or 12. Since you’re describing the volume of a 3-dimensional object, remember to write your answer in cubic units. In this case, the pyramid has a volume of 12 cubic inches. For pyramids with a triangular base, the technique is a little different. If you know the triangle’s height and the width of its base, plug those numbers into the formula ½ x b (base) x h (height) to find the area of the triangle. From there, you can use the same formula that you used for the square-based pyramid. For example, say your pyramid has a base that’s a triangle with a base width of 2 cm and a height of 4 cm, and the pyramid has a height of 6 cm. First, find the area of the triangle using the formula ½ x 2 x 4, which will give you a base area of 4 square centimeters. Next, multiply the area by the height of the pyramid, then multiply the product by 1/3. In this case, 6 x 4 x 1/3 = 8, which means the pyramid has a volume of 8 cubic centimeters. To learn how to calculate the volume of a pyramid with a triangular base, read on!
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