# Surface Area of a Trapezoidal Prism

The equation for the surface area of a trapezoidal prism is a little more complicated than the formula for the volume of a trapezoidal prism, but as long as you know the length of the prism, the height of the trapezoid cross-sections, and the base and top lengths of the trapezoid, you can easily work out the area of the surface. A trapezoidal prism has six faces and the total surface area is simply the sum of the areas of these six faces. Four of the faces are rectangles and two are trapezoids.

This tutorial will show you the equation for trapezoid prism surface area in two cases: prisms with symmetric trapezoid cross-sections, and prisms with asymmetric cross-sections. Each is accompanied by examples to help you see the calculations in action. Finding the surface area is useful in constructing a box, storage container, or feeding trough in the shape of a trapezoidal prism. See also, **Volume Formula for a Trapezoidal Prism**

## Surface Area Formula for Symmetric Trapezoidal Prisms

If the top and bottom of the trapezoid are aligned so that their midpoints lie on the same vertical axis, then we can use the symmetric surface area formula. Let H be the height of the prism, L be the prism length, B be the trapezoid's base width, and A be the trapezoid's top width. Then the sides of the trapezoid are of equal length, S. This side length is given by the equation

S = sqrt[ H^2 + 0.25(B-A)^2 ]

This comes from applying the Pythagorean Theorem to a triangle with legs of length H and (B-A)/2. The height can also be expressed in terms of the slant slide length,

H = sqrt[ S^2 - 0.25(B-A)^2 ]

The total surface area of the four rectangular faces is

(B + A + 2S)*L

The total surface area of the two trapezoidal faces is

2*[H*(B+A)/2] = H*(B + A) = (B+A)*sqrt[ S^2 - 0.25(B-A)^2 ]

Therefore, the total surface area of the symmetric trapezoidal prism is

**Surface Area ****= (B + A + 2S)*L + (B + A)*H****= (B+A)*(L+H) + L*sqrt[ 4H^2 + (B-A)^2 ]= (B + A + 2S)*L + (B + A)*sqrt[ S^2 - 0.25(B-A)^2 ]**

## Example Surface Area Problem #1

Suppose you have a symmetric trapezoidal prism with H = 3, L = 8, B = 7, and A = 4 as in the diagram above. Since the edges B and A are centered, the value of S is given by

S = sqrt[ H^2 + 0.25(B-A)^2 ]

= sqrt[ 9 + 0.25*9 ]

= sqrt[ 11.25 ]

≈ 3.3541

Plugging the values of H, L, B, A, and S into the surface area formula gives us

Surface Area

= (B+A)(L+H) + 2LS

= (7+4)(8+3) + 2*8*3.3541

= 11*11 + 16*3.3541

= 174.6656

## Surface Area Formula for Asymmetric Trapezoidal Prisms

If the top and bottom of the trapezoid are not aligned so that their midpoints lie on the same vertical axis, then we must use the ssymmetric surface area formula. As before, let H be the height of the prism, L be the prism length, B be the trapezoid's base width, and A be the trapezoid's top width. Also let S_{1} be the length of one side of the trapezoid and let S_{2} be the length of the other. The total surface area of the four rectangular faces of the trapezoidal prism is

(B + A + S_{1} + S_{2})*L

The total surface area of the two trapezoidal faces is

2*[H*(B+A)/2] = H*(B + A)

which is the same as for the symmetric case. Therefore, the total surface area of the symmetric trapezoidal prism is

**Surface Area ****= (B + A + S _{1} + S_{2})*L + (B + A)*H**

**= (B+A)*(L+H) + L*(S**

_{1}+ S_{2})## Example Surface Area Problem #2

Consider an asymmetric trapezoidal prism whose length is 6.03 cm, height 3.63 cm, base with width 7.82 cm, top width 3.07 cm, and whose two trapezoidal edge lengths are 4.03 cm and 4.71 cm. Thus we have L = 6.03, H = 3.63, B = 7.82, A = 3.07, S_{1} = 4.03, and S_{2} = 4.71.

The surface ares of the shape is found be plugging these six measurements into the formula given above.

Surface Area

= (B+A)(L+H) + L*(S_{1} + S_{2})

= (7.82+3.07)(6.03+3.63) + 6.03(4.03+4.71)

= 10.89*9.66 + 6.03*8.74

= 157.8996

## Example Surface Area Problem #3

A symmetric trapezoidal prism has a length of 25 inches, height of 6 inches, and top width of 11 inches. It's base width is unknown, but it has a total surface area of 1678 square inches. Can you determine the width of the trapezoid base from this information?

First, we have L = 25, H = 6, T = 11, and surface area = 1678. Since the trapezoidal cross-sections are symmetric trapezoids, we can use the first surface area formula,

Surface Area = (L+H)(B+A) + L*sqrt[ 4H^2 + (B-A)^2 ]

Filling in the known information gives us

1678 = (25+6)(B+11) + 25*sqrt[ 4*6^2 + (B-11)^2 ]

1678 = 31B + 341 + 25*sqrt[ B^2 - 22B + 265 ]

1337 - 31B = 25*sqrt[B^2 - 22B + 265]

At this point, the best way to solve for B is to square both sides and group like terms to produce a quadratic equation. Then we can either factor it or use the quadratic formula to find B.

1787569 - 82894B + 961B^2 = 625B^2 - 13750B + 165625

336B^2 - 69144B + 1621944 = 0

14B^2 - 2881B + 67581 = 0

(B - 27)(14B - 2503) = 0

B = 27 or B = 2503/14 = 178.786

Plugging B = 27 into the surface area formula yields the correct answer of 1678. Plugging B = 178.786 into the formula yields a different answer, so we can discard this solution. Therefore, the unique solution is B = 27 inches.

## Calculus and Geometry Optimization Problem: Find the Minimum Surface Area

A trapezoidal prism has a volume of 1350. If the trapezium face is symmetric with angles of 60 degrees and 120 degrees, what dimensions of the trapezoidal prism yield the minimal surface area?

**Solution:** The trapezoidal cross-sections of the prism look like the image below. This is the trapezoid formed by attaching three equal sized equilateral triangles together.

Letting the shorter parallel side be x, the lengths of the slant sides are also x and the length of the longer parallel side is 2x. The height of the trapezoid is (sqrt(3)/2)x. If we call the length of the trapezoidal prism L, then the volume formula gives us

1350 = L(x+2x)(x)(sqrt(3)/4)

= (3sqrt(3)/4)Lx^2

Solving for L gives us

L = 600sqrt(3)/x^2

Using the surface area formula for trapezoidal prisms, we get the equation

Surface Area = 2(x+2x)(x)sqrt(3)/4 + 5x*L

= (3sqrt(3)/2)x^2 + 3000sqrt(3)/x

If we call this expression f(x), then we have

f'(x) = 3sqrt(3)x - 3000sqrt(3)/x^2

Setting the derivative equal to zero and solving for x gives us

3000sqrt(3)/x^2 = 3sqrt(3)x

1000 = x^3

x = 10

This is only critical point of the function f(x) = (3sqrt(3)/2)x^2 + 3000sqrt(3)/x, pictured in the graph below, therefore, the minimum surface area is obtained by making the trapezoidal face have side lengths of 10, 10, 10, and 20, and making the length equal to 6*sqrt(3) ≈ 10.3923. These dimensions yield a trapezoidal prism with a surface area of 450sqrt(3) ≈ 779.4229

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