How to Calculate the Volume of a Triangular Prism: Formulas & Examples

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

Find the area of the triangular base with $Area={\frac {1}{2}}bh$ .
Identify the height of the prism (the distance between the 2 triangular faces).
Enter the base area $B$ and the height $H$ into $Volume=BH$ .
Solve the equation to find the total volume of the triangular prism.

Section 1 of 4:

Formula for the Volume of a Triangular Prism

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Use the formula $V=BH$ 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, $V$ is the total volume of the prism, $B$ is the area of one of the triangular bases, and $H$ is the height of the prism (the distance between the triangular bases, not the height of the triangular base itself). ^{[1]
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- You may also see the equation written as $V=({\frac {1}{2}}bh)H$ , where $b$ and $h$ represent the base and height of the triangular base. This is because ${\frac {1}{2}}bh$ is the formula for the area of a triangle.

Section 2 of 4:

Finding the Volume of a Triangular Prism

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1
Use $Area={\frac {1}{2}}bh$ to find the area of the triangular base. Say the triangle has a base $b$ of 8 cm and a height $h$ of 9 cm. Just plug those values into $A={\frac {1}{2}}bh$ to find the area of the base: ^{[2]
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- $A={\frac {1}{2}}*8*9=36$ . The area of the triangle is 36 cm ² .
- Insert 36 for $B$ in the volume formula $V=BH$ .
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 $H$ place of the formula $V=BH$ . ^{[3]
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- For example, your formula should now look like $V=36*16$ .
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]
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- So, if $V=36*16$ , the answer is 576 cm ³ .
- Remember that when finding volume, the answer will be in cubed units (and area is in square units).

Section 3 of 4:

Finding the Area of the Base Triangle without a Given Height

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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]
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Research source} If you know the length of the hypotenuse ( $c$ ) and only one leg ( $a$ or $b$ ), plug those values into the Pythagorean Theorem ( a ² + b ² = c ² ) to find the length of the missing leg: ^{[6]
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Research source} Then, you can calculate the area of the triangle . ^{[7]
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- Say the hypotenuse is 6 units and the base $a$ is 4 units:
  - 4 ² + b ² = 6 ²
  - 16 + b ² = 36
  - b ² = 20
  - b = √20 (exact answer) or 4.472 (rounded answer).
- Once you know the height, you can use $A={\frac {1}{2}}bh$ to find the triangle’s area:
  - A = ½*4*√20 = 2√20 (exact answer) or 8.944 (rounded answer)
2
Isosceles triangle: A = ½[√(a ² - b ² /4) x b] In this case, $a$ is the length of both equal sides and $b$ is the length of the third, unequal side. ^{[8]
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Research source} So, let’s say the equal sides ( $a$ ) have a value of 4 and the base ( $b$ ) has a value of 2:
- A = ½[√(4 ² - 2 ² /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 ( $a$ becomes the hypotenuse and $b/2$ becomes one leg). Then, use the Pythagorean Theorem to solve for the height (the other leg).
3
Equilateral triangle: A = (a ² 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 ( $a$ ) into the formula: ^{[9]
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Research source} Say $a$ has a value of 6:
- A = (6 ² x √3)/4
- A = (36 x √3)/4
- A = 9√3 (exact answer) or 15.588 (rounded answer)
4
Scalene triangle: A = √s(s - a)(s - b)(s - c) This is known as Heron’s formula. In it, $s$ is half of the perimeter of the triangle (or ( a + b + c )/2 ) and $a$ , $b$ , and $c$ are the lengths of the 3 sides. ^{[10]
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Research source} So, let’s say you have a triangle with sides that measure 2, 3, and 4 units long:
- First, find the value of $s$ :
  - 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)

Section 4 of 4:

Properties of Triangular Prisms

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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]
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- 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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How do I calculate the area of a triangular prism?

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To find the surface area of a triangular prism, you would find the area of all the faces, then add them together.

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How can I find the volume of a rectangular prism?

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To find the volume of a rectangular prism, multiply the prism's width, height, and length. V = w*h*l

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How do I find the volume of a square pyramid?

Donagan

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V = (1/3)(s²)(h), where s is the length of a side of the base, and h is the height.

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The formula $V=BH$ can be used to find the volume of any prism.

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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.

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How to Calculate the Volume of a Triangular Prism: Formulas & Examples

Calculating Triangular Prism Volume Step by Step

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Formula for the Volume of a Triangular Prism

Finding the Volume of a Triangular Prism

Finding the Area of the Base Triangle without a Given Height

Properties of Triangular Prisms

Triangular Prism Volume Calculator, Practice Problems, and Answers

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