Volume of a Trapezoidal Prism: Formula and Examples
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.
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
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.
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.