# Volume of a Trapezoidal Prism: Formula and Examples

TR Smith is a product designer and former teacher who uses math in her work every day.

In geometry, a trapezoidal prism is a solid shape that has trapezoid (or trapezium in the UK) cross-sections in one direction and rectangular cross-sections in the other directions. To compute the volume of a symmetric trapezoidal prism, you need to know four measurements: the length of the prism L, the height of the trapezoidal cross-section H, the base width of the trapezoid B, and the top width of the trapezoid A.

Alternatively, if you know the trapezoid's slant side lengths S, you can compute the volume with L, S, B, and A.

Both formulas for the volume of a trapezoidal pyramid are given below along with several example problems. See also, Surface Area Formula for a Trapezoidal Prism.

## Formula for Volume of a Trapezoidal Prism

If the prism length is L, trapezoid base width B, trapezoid top width A, and trapezoid height H, then the volume of the prism is given by the four-variable formula:

V(L, B, A, H) = LH(A + B)/2.

In other words, multiply together the length, height, and average of A and B. This formula is equivalent to multiplying the length of the prism by the area of the trapezoidal cross-sections. If you don't know H, but instead know the slant side length S, the formula is a bit more complicated. It is:

V(L, B, A, S) = L(A + B)sqrt(4S2 + 2AB - B2 - A2)/4.

This second formula is derived from the fact that:

H = sqrt[S2 - ((B - A)/2)2]
= sqrt(4S2 + 2AB - B2 - A2)/2.

Here are some example problems to help you work out prism volumes. In the formulas above and examples below, it is assumed that the trapeziums are symmetric—that is, the slant side lengths are equal on both sides and the center of the top length is vertically aligned with the center of the base length.

## Example 1

A trapezoidal prism has a length of 8, base width of 7, top width of 4, and height of 3.

Using H = 3, B = 7, A = 4, and L = 8, we can compute the volume using the first equation. Plugging the variables into that equation gives us:

V = LH(A + B)/2
= 8*3*(4 + 7)/2
= 24*11/2
= 132.

## Example 2

A trapezoidal pyramid has length of 6.03 cm. Its base width is 7.82 cm, top width 3.55 cm, and slant side length 4.71 cm. What is its volume in cubic centimeters?

This problem gives us L = 6.03, B = 7.82, A = 3.55, and S = 4.71. Since we have S instead of H, we use the second volume equation. Plugging in these values gives us:

V = L(A + B)sqrt(4S^2 + 2AB - B^2 - A^2)/4
= 6.03(3.55 + 7.82)sqrt(4*4.71^2 + 2*3.55*7.82 - 7.82^2 - 3.55^2)/4
= 143.92 cm^3.

So the volume is 143.92 cubic centimeters, or equivalently 0.14392 liters.

## Example 3

A trapezoidal prism has a volume of 1950 cubic inches. The height is 6 inches, the length is 25 inches, and the top width is 11 inches. What is the width of the trapezoid base?

Here we are given the volume but one of the measurements is missing. We have V = 1950, H = 6, L = 25, and A = 11, but B is unknown. Therefore we need to plug these values into the first volume formula and solve for B. This gives us:

V = LH(A + B)/2
1950 = 25*6*(11 + B)/2
1950/(25*6) = (11 + B)/2
13 = (11 + B)/2
2*13 = 11 + B
26 = 11 + B
26 - 11 = B
15 = B

So the width of the base is 15 inches.

## Calculus Optimization Example: Maximize Volume of Trapezium Prism with Given Length and Base

Suppose you want to make a box in the shape of a trapezoidal prism subject to these three conditions: The length of the prism box must be 24 cm, one of the parallel sides of the trapezoid face must be 12 cm long, and the whole perimeter of the trapezoidal face must be 34 cm. What shape should the trapezium be so that the volume of the box is maximized?

To begin, we should note that this is essentially a problem about maximizing the area of the trapezoid cross-sections since the length is constant at 24 cm.

To come up the equations we need to optimize with calculus, first let x equal the length of each slant side. Since one of the parallel sides is 12, the other must be 34 - 12 - x - x, or 22 - 2x. Below are the possible trapezoid shapes that fit the constraints of the problem; the triangle and flat line are the limiting cases.

Regardless of the shape of the trapezoid, its height can be found by applying the Pyathagorean Theorem to the right triangle formed on the trapezoid's side. Simplifying the expression

h^2 + |x - 5|^2 = x^2

gives us h = sqrt(10x - 25). See figure above. Since the area of a trapezoid is 1/2 times height times the sum of the two parallel side lengths, we have:

Area = (1/2)*sqrt(10x - 25)*(34 - 2x) = (17 - x)*sqrt(10x - 25).

To maximize the area, we take the derivative of the function, set it equal to zero, and solve for x. Using the product rule, the derivative is:

[ (17 - x)*sqrt(10x - 25) ]' = (110x - 15)/sqrt(10x - 25).

This derivative equals 0 when x = 110/15 = 22/3, or about 7.33333. Plugging this value into the area function gives us a maximum trapezoidal area of:

(17 - 22/3)*sqrt(220/3 - 25) = (29/3)*sqrt(145/3) ≈ 67.20477 cm^2.

The maximum volume of the trapezoidal prism is then 24*(29/3)*sqrt(145/3) ≈ 1612.91455 cm^3. The solution prism is shown below.

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• Gerald 2 years ago

Thanks, this was very useful. If I want to make a trapezoidal prism with a given length, a given side length for one of the trapezoid bases, and a given total perimeter for the trapezoid faces, what should the lengths of the other sides of the trapezium be to maximize the area, and therefore the volume of the prism?

• Author

TR Smith 2 years ago

Hi Gerald,

If the perimeter is fixed at P and one of the parallel side lengths is fixed at X, then to maximize the area of the trapezoid, the other three side lengths should be equal to (P-X)/3.

• dan 2 years ago

helpful to know this formula, i am trying to make some specialty boxes that are not a rectangular shape but can still be packed without gaps. the trapezoidal prism seems like a good choice. are there any other 3d shapes that can 'tessellate' in 3d space?

• Author

TR Smith 2 years ago

Hi Dan,

Prisms whose cross-sections are triangles or certain types of quadrilaterals and hexagons will tessellate in 3D space, as will some types of pentagonal prisms. Going beyond prisms, a rhombic dodedecahedron is another space-filling polyhedron. (See https://owlcation.com/stem/Names-of-Geometric-Shap... )

• celia 2 years ago

helpful to have the formula since i am writing a calculator for prism volumes of various shapes. user picks triangle, rectangle, trapezoid, or hexagon and enters the measurements according to the shape, output is the volume and surface area.

• Kara 22 months ago

• Sindan 22 months ago

If I have a trapezoidal prism and want the volume to increase by a factor of 1.8 by how much do i need to increase the side lengths?

• Author

TR Smith 22 months ago

If you want the trapezoidal prism to maintain the same proportions, then you need to increase each linear dimension by a factor of (1.8)^(1/3), which is approximately 1.21644.

• Sagar gupta 21 months ago

Thanks! This is really very useful.

• Abdiwahab Wadar Amey 11 months ago

How can we defferentiate the volume of a trapezoidal prism from the volume of Frustam of pyramid, Since they have similar shapes.

• Author

TR Smith 11 months ago

Hi AWA, thanks for the question. The trapezoidal prism and square pyramidal frustum are similar but not the same. The trapezoidal prism has two pairs of parallel faces while the square pyramidal frustum has only one pair of parallel faces. Since the latter has more slanting faces with respect to the base than the former, their volumes are computed differently.

• Needs Help 10 months ago

What about a trapezoidal prism w/one right angle? Would the formula be the same? I would just like the formula. I used the one you put on here, but keep getting the answer wrong. (And I used a calculator) :/

• Author

TR Smith 9 months ago

Hi NH, thanks for the question. The formula isn't working for you because the shape you are dealing with isn't a trapezoid. Since a trapezoid is defined to have exactly one pair of parallel sides it can have either 0 or 2 right angles. It cannot have exactly 1 right angle.

With out a diagram of the shape you are working with I can't give you a formula for the volume. Kites are a family of symmetric quadrilaterals that could have exactly 1 right angle. If you're talking about a kite prism, then the volume is the area of the kite times the length of the prism. The area of a kite is one half of the product of the diagonal lengths.