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Terms of Use wikiHow is where trusted research and expert knowledge come together. Learn why people trust wikiHow How to Calculate the Volume of a Triangular Prism: Formulas & Examples PDF download Download Article Plus, how to find the area of a triangle when the height is unknown Co-authored by Grace Imson, MA and Dan Hickey

Last Updated: October 24, 2024 Fact Checked

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  • Triangular Prism Volume Formula
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  • Finding the Volume of a Triangular Prism
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  • Finding Base Area without a Given Height
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  • Properties of Triangular Prisms
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This article was co-authored by Grace Imson, MA and by wikiHow staff writer, Dan Hickey. 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. There are 10 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,294,753 times.

Finding the volume of a triangular prism is easier than you might think. All you have to do is find the area of one of the triangular bases, then multiply it by the height of the prism! In this article, we’ll show you how to do just that, including how to find the area of the base whether the height is given or not. We’ll also review more properties of triangular prisms so you’re fully prepared for your next geometry exam.

Calculating Triangular Prism Volume Step by Step

  1. Find the area of the triangular base with A r e a = 1 2 b h {\displaystyle Area={\frac {1}{2}}bh} .
  2. Identify the height of the prism (the distance between the 2 triangular faces).
  3. Enter the base area B {\displaystyle B} and the height H {\displaystyle H} into V o l u m e = B H {\displaystyle Volume=BH} .
  4. Solve the equation to find the total volume of the triangular prism.

Steps

Section 1 of 4:

Formula for the Volume of a Triangular Prism

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  1. Use the formula... {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/e\/e0\/Calculate-the-Volume-of-a-Triangular-Prism-Step-1-Version-4.jpg\/v4-460px-Calculate-the-Volume-of-a-Triangular-Prism-Step-1-Version-4.jpg","bigUrl":"\/images\/thumb\/e\/e0\/Calculate-the-Volume-of-a-Triangular-Prism-Step-1-Version-4.jpg\/v4-728px-Calculate-the-Volume-of-a-Triangular-Prism-Step-1-Version-4.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is <b>not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc.<br>\n<\/p><p><br \/>\n<\/p><\/div>"} Use the formula V = B H {\displaystyle V=BH} to find the volume of a triangular prism. According to physics teacher Joseph Quinones, “The volume of a triangular prism can be calculated by multiplying the base area of the triangle by the height of the prism.” In this formula, V {\displaystyle V} is the total volume of the prism, B {\displaystyle B} is the area of one of the triangular bases, and H {\displaystyle H} is the height of the prism (the distance between the triangular bases, not the height of the triangular base itself).[1]
    • You may also see the equation written as V = ( 1 2 b h ) H {\displaystyle V=({\frac {1}{2}}bh)H} , where b {\displaystyle b} and h {\displaystyle h} represent the base and height of the triangular base. This is because 1 2 b h {\displaystyle {\frac {1}{2}}bh} is the formula for the area of a triangle.
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Section 2 of 4:

Finding the Volume of a Triangular Prism

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  1. Step 1 Use ... {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/9\/9c\/Calculate-the-Volume-of-a-Triangular-Prism-Step-1-Version-3.jpg\/v4-460px-Calculate-the-Volume-of-a-Triangular-Prism-Step-1-Version-3.jpg","bigUrl":"\/images\/thumb\/9\/9c\/Calculate-the-Volume-of-a-Triangular-Prism-Step-1-Version-3.jpg\/v4-728px-Calculate-the-Volume-of-a-Triangular-Prism-Step-1-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"} 1 Use A r e a = 1 2 b h {\displaystyle Area={\frac {1}{2}}bh} to find the area of the triangular base. Say the triangle has a base b {\displaystyle b} of 8 cm and a height h {\displaystyle h} of 9 cm. Just plug those values into A = 1 2 b h {\displaystyle A={\frac {1}{2}}bh} to find the area of the base:[2]
    • A = 1 2 ∗ 8 ∗ 9 = 36 {\displaystyle A={\frac {1}{2}}*8*9=36} . The area of the triangle is 36 cm 2.
    • Insert 36 for B {\displaystyle B} in the volume formula V = B H {\displaystyle V=BH} .
  2. Step 2 Identify the height of the prism. {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/3\/3c\/Calculate-the-Volume-of-a-Triangular-Prism-Step-5.jpg\/v4-460px-Calculate-the-Volume-of-a-Triangular-Prism-Step-5.jpg","bigUrl":"\/images\/thumb\/3\/3c\/Calculate-the-Volume-of-a-Triangular-Prism-Step-5.jpg\/v4-728px-Calculate-the-Volume-of-a-Triangular-Prism-Step-5.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"} 2 Identify the height of the prism. Now you need to find the height of the triangular prism, which is the length of 1 of its sides (this is different from the height of the triangular base). For example, the prism may be 16 cm high. Place this number in the H {\displaystyle H} place of the formula V = B H {\displaystyle V=BH} .[3]
    • For example, your formula should now look like V = 36 ∗ 16 {\displaystyle V=36*16} .
  3. Step 3 Multiply the base area by the height of the prism to find the volume. {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/8\/89\/Calculate-the-Volume-of-a-Triangular-Prism-Step-6.jpg\/v4-460px-Calculate-the-Volume-of-a-Triangular-Prism-Step-6.jpg","bigUrl":"\/images\/thumb\/8\/89\/Calculate-the-Volume-of-a-Triangular-Prism-Step-6.jpg\/v4-728px-Calculate-the-Volume-of-a-Triangular-Prism-Step-6.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"} 3 Multiply the base area by the height of the prism to find the volume. Since you now have all the parts of the equation, multiply the area of the triangular base by the height of the prism. The result will be the volume of the triangular prism.[4]
    • So, if V = 36 ∗ 16 {\displaystyle V=36*16} , the answer is 576 cm 3.
    • Remember that when finding volume, the answer will be in cubed units (and area is in square units).
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Section 3 of 4:

Finding the Area of the Base Triangle without a Given Height

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  1. Step 1 Right triangle: Pythagorean Theorem {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/a\/a4\/Calculate-the-Volume-of-a-Triangular-Prism-Step-5-Version-2.jpg\/v4-460px-Calculate-the-Volume-of-a-Triangular-Prism-Step-5-Version-2.jpg","bigUrl":"\/images\/thumb\/a\/a4\/Calculate-the-Volume-of-a-Triangular-Prism-Step-5-Version-2.jpg\/v4-728px-Calculate-the-Volume-of-a-Triangular-Prism-Step-5-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is <b>not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc.<br>\n<\/p><p><br \/>\n<\/p><\/div>"} 1 Right triangle: Pythagorean Theorem A right triangle is a triangle where one angle equals 90° (a right angle); the side opposite the right angle is the hypotenuse and the other 2 sides are the legs (one is the base and one is the height, but they can be labeled interchangeably).[5] If you know the length of the hypotenuse ( c {\displaystyle c} ) and only one leg ( a {\displaystyle a} or b {\displaystyle b} ), plug those values into the Pythagorean Theorem (a2 + b2 = c2) to find the length of the missing leg:[6] Then, you can calculate the area of the triangle.[7]
    • Say the hypotenuse is 6 units and the base a {\displaystyle a} is 4 units:
      • 42 + b2 = 62
      • 16 + b2 = 36
      • b2 = 20
      • b = √20 (exact answer) or 4.472 (rounded answer).
    • Once you know the height, you can use A = 1 2 b h {\displaystyle A={\frac {1}{2}}bh} to find the triangle’s area:
      • A = ½*4*√20 = 2√20 (exact answer) or 8.944 (rounded answer)
  2. Step 2 Isosceles triangle: A = ½[√(a2 - b2/4) x b] {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/3\/3e\/Calculate-the-Volume-of-a-Triangular-Prism-Step-6-Version-2.jpg\/v4-460px-Calculate-the-Volume-of-a-Triangular-Prism-Step-6-Version-2.jpg","bigUrl":"\/images\/thumb\/3\/3e\/Calculate-the-Volume-of-a-Triangular-Prism-Step-6-Version-2.jpg\/v4-728px-Calculate-the-Volume-of-a-Triangular-Prism-Step-6-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is <b>not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc.<br>\n<\/p><p><br \/>\n<\/p><\/div>"} 2 Isosceles triangle: A = ½[√(a2 - b2/4) x b] In this case, a {\displaystyle a} is the length of both equal sides and b {\displaystyle b} is the length of the third, unequal side.[8] So, let’s say the equal sides ( a {\displaystyle a} ) have a value of 4 and the base ( b {\displaystyle b} ) has a value of 2:
    • A = ½[√(42 - 22/4) x 2]
    • A = ½[√(16 - 4/4) x 2]
    • A = ½[√(16 - 1) x 2]
    • A = ½[√15 x 2]
    • A = √15 (exact answer) or 3.873 (rounded answer)
    • Alternate method: Divide your triangle in 2 vertically to form a right triangle ( a {\displaystyle a} becomes the hypotenuse and b / 2 {\displaystyle b/2} becomes one leg). Then, use the Pythagorean Theorem to solve for the height (the other leg).
  3. Step 3 Equilateral triangle: A = (a2 x √3)/4 {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/1\/16\/Calculate-the-Volume-of-a-Triangular-Prism-Step-7.jpg\/v4-460px-Calculate-the-Volume-of-a-Triangular-Prism-Step-7.jpg","bigUrl":"\/images\/thumb\/1\/16\/Calculate-the-Volume-of-a-Triangular-Prism-Step-7.jpg\/v4-728px-Calculate-the-Volume-of-a-Triangular-Prism-Step-7.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is <b>not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc.<br>\n<\/p><p><br \/>\n<\/p><\/div>"} 3 Equilateral triangle: A = (a2 x √3)/4 An equilateral triangle is a triangle where all sides are equal in length. To find the area (when the height is unknown), just plug in the length of one of the sides ( a {\displaystyle a} ) into the formula:[9] Say a {\displaystyle a} has a value of 6:
    • A = (62 x √3)/4
    • A = (36 x √3)/4
    • A = 9√3 (exact answer) or 15.588 (rounded answer)
  4. Step 4 Scalene triangle: A = √s(s - a)(s - b)(s - c) {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/5\/5b\/Calculate-the-Volume-of-a-Triangular-Prism-Step-8.jpg\/v4-460px-Calculate-the-Volume-of-a-Triangular-Prism-Step-8.jpg","bigUrl":"\/images\/thumb\/5\/5b\/Calculate-the-Volume-of-a-Triangular-Prism-Step-8.jpg\/v4-728px-Calculate-the-Volume-of-a-Triangular-Prism-Step-8.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is <b>not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc.<br>\n<\/p><p><br \/>\n<\/p><\/div>"} 4 Scalene triangle: A = √s(s - a)(s - b)(s - c) This is known as Heron’s formula. In it, s {\displaystyle s} is half of the perimeter of the triangle (or (a+b+c)/2) and a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are the lengths of the 3 sides.[10] So, let’s say you have a triangle with sides that measure 2, 3, and 4 units long:
    • First, find the value of s {\displaystyle s} :
      • s = (2 + 3 + 4)/2
      • s = 9/2 = 4.5
    • Then, plug in all values to find the area:
      • A = √4.5(4.5 - 2)(4.5 - 3)(4.5 - 4)
      • A = √4.5(2.5)(1.5)(0.5)
      • A = √4.5(2.5)(1.5)(0.5)
      • A = √4.5(1.875)
      • A = √8.4375 (exact answer) or 2.905 (rounded answer)
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Section 4 of 4:

Properties of Triangular Prisms

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  1. Triangular prisms have 2 triangular bases and 3 parallelogram faces. {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/8\/82\/Calculate-the-Volume-of-a-Triangular-Prism-Step-9.jpg\/v4-460px-Calculate-the-Volume-of-a-Triangular-Prism-Step-9.jpg","bigUrl":"\/images\/thumb\/8\/82\/Calculate-the-Volume-of-a-Triangular-Prism-Step-9.jpg\/v4-728px-Calculate-the-Volume-of-a-Triangular-Prism-Step-9.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>\u00a9 2024 wikiHow, Inc. All rights reserved. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. This image is <b>not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image may not be used by other entities without the express written consent of wikiHow, Inc.<br>\n<\/p><p><br \/>\n<\/p><\/div>"} Triangular prisms have 2 triangular bases and 3 parallelogram faces. Prisms are named for the shape of their base, so the base of any triangular prism is a triangle. The base has a congruent (same size and shape) and parallel face that is also a triangle at the other end of the prism. Since a triangle has 3 sides, that means it takes 3 more faces to fully enclose the prism. These faces are parallelograms (polygons with 2 sets of parallel sides, like rectangles or squares) that connect the triangles together. The base of each parallelogram is equal to the length of the triangle side it connects to, and the height of each parallelogram is the total height of the prism.[11]
    • Every cross-section of the prism parallel to the base will also be a congruent triangle.
    • Triangular prisms have 5 faces, 6 vertices, and 9 edges.
    • A triangular prism is also known as a pentahedron because it has 5 faces (penta means 5).

Triangular Prism Volume Calculator, Practice Problems, and Answers

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  • Question How do I calculate the area of a triangular prism? Community Answer Community Answer To find the surface area of a triangular prism, you would find the area of all the faces, then add them together. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 270 Helpful 341
  • Question How can I find the volume of a rectangular prism? Community Answer Community Answer To find the volume of a rectangular prism, multiply the prism's width, height, and length. V = w*h*l Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 184 Helpful 230
  • Question How do I find the volume of a square pyramid? Donagan Donagan Top Answerer V = (1/3)(s²)(h), where s is the length of a side of the base, and h is the height. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 157 Helpful 97
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  • The formula V = B H {\displaystyle V=BH} can be used to find the volume of any prism. Thanks Helpful 0 Not Helpful 0
  • Ensure that the units of measurement are the same for all parts of the triangular prism before you begin the calculation. For example, if 1 part of the prism is in millimeters and the rest is in centimeters, convert the millimeters to centimeters first. Thanks Helpful 0 Not Helpful 0
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Expert Interview

Thanks for reading our article! If you’d like to learn more about physics, check out our in-depth interview with Joseph Quinones.

References

  1. ↑ https://flexbooks.ck12.org/cbook/interactive-math-7-flexlet/section/4.6/related/lesson/volume-of-triangular-prisms-msm7/
  2. ↑ https://flexbooks.ck12.org/cbook/ck-12-cbse-maths-class-6/section/13.6/primary/lesson/area-of-triangle/
  3. ↑ https://www.ck12.org/geometry/volume-of-triangular-prisms/lesson/Volume-of-Triangular-Prisms-MSM7/
  4. ↑ https://flexbooks.ck12.org/cbook/ck-12-interactive-middle-school-math-7-for-ccss/section/6.10/related/lesson/volume-of-triangular-prisms-msm7/
  5. ↑ https://www.palmbeachstate.edu/prepmathlw/Documents/the_pythagorean_theorem.pdf
  6. ↑ https://www.palmbeachstate.edu/prepmathlw/Documents/the_pythagorean_theorem.pdf
  7. ↑ https://content.byui.edu/file/b8b83119-9acc-4a7b-bc84-efacf9043998/1/Math-2-4-6.html
  8. ↑ https://www.cuemath.com/measurement/area-of-an-isosceles-triangle/
  9. ↑ https://www.omnicalculator.com/math/equilateral-triangle
More References (2)
  1. ↑ https://www.cuemath.com/measurement/area-of-scalene-triangle/
  2. ↑ https://amsi.org.au/ESA_middle_years/Year6/Year6_2cT/Year6_2cT_R1_pg3.html

About This Article

Grace Imson, MA Co-authored by: Grace Imson, MA Math Teacher This article was co-authored by Grace Imson, MA and by wikiHow staff writer, Dan Hickey. 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 1,294,753 times. How helpful is this? Co-authors: 70 Updated: October 24, 2024 Views: 1,294,753 Categories: Volume Article SummaryX

To calculate the volume of a triangular prism, first you need to find the area of one of the triangular bases by multiplying ½ by the base of the triangle and by the height of the triangle. For example, if the base is 8 and the height is 9, you would get ½ x 8 x 9 = 36. Therefore, the area is 36. Next, plug the area into the formula for finding the volume of a triangular prism, which is V = b x h, or volume equals the area of the base multiplied by the height of the prism. Now, plug the height of the prism into the formula and solve. As an example, if the area of the triangular base is 36 and the height of the prism is 16, you would get 36 x 16 = 576. Therefore, the volume of the triangular prism is 576. Write your answer in cubic units. If you want to learn how to calculate the height of a regular triangle, keep reading! Did this summary help you?YesNo

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