# How to Find the Missing Side of a Trapezoid Given the Area

It's easy to find the missing side or sides of a trapezoid given two or three sides and the trapezoid's perimeter. However, it is more difficult to find the missing side of a trapezoid if you know the area and two or three sides.

For example, if an asymmetric trapezoid has a perimeter of 86 feet and three of its sides are 40, 19, and 10 feet long, then the missing side must be 86 - 40 - 19 - 10 = 17 feet. Or if a symmetric trapezoid has a perimeter of 86 and two of its sides are 40 and 20, then the missing sides are either {13, 13}, or {20, 6}. This problem is easy to solve using nothing more than basic arithmetic.

But how do you solve for the side length of a trapezoid if you know the area is 236 and three of the sides are 40, 19, and 10? And how does the answer change depending on whether the trapezoid is symmetric or asymmetric, or whether the missing side is a parallel side or slant side? This question is harder, but still possible to answer using geometric properties of trapezoids and algebra. Below are some examples to show how to solve for the missing sides of a trapezoid.

## Example 1: Symmetric Trapezoid, 4 Solutions

A symmetric trapezoid has an area of 200 and two of its sides are 30 and 18. What could be the lengths of the two missing sides?

When you are only given two sides of a symmetric trapezoid, there are several possible solutions depending on whether you assume the given sides are parallel, or whether one of them is a slant side. Here are the three possibilities for this problem:

- Case I: 30 and 18 are parallel, missing sides are slanted
- Case II: 30 is parallel, slanted sides are 18, missing side is parallel
- Case III: 18 is parallel, slanted sides are 30, missing side is parallel

Note that we cannot have 18 and 30 as the two slant sides, since this contradicts the statement of the problem, that the trapezoid is symmetric. Let's analyze the three cases to determine what the set of sides can be.

**Case I:** If the parallel sides are 30 and 18 and the area is 200, then we can first find the height of the trapezoid using the base-height trapezoidal area formula, Area = H(P+Q)/2, where H is the height and P and Q are the lengths of the parallel sides. Using P = 30, Q = 18, and Area = 200, we get

200 = H(30+18)/2

H = 400/48 = 25/3

Now with the height, we can use the Pythagorean Theorem to find the lengths of the slant sides.

As you can see from the image above, the length of the slant sides is the hypotenuse of a right triangle with legs 6 and 25/3. This means the slanted side length is sqrt[6^2 + (25/3)^2] ≈ 10.2686. Thus the solution for the side lengths in Case I is {30, 18, 10.2686, 10.2686} with 30 and 18 as the two parallel sides and 10.2686 as the length of the slant sides.

**Case II:** If 30 is the length of a parallel side and 18 is the length of a slanted side, let's call the missing parallel side X. As before, we first find the height H using the base-height area formula.

200 = H(30+X)/2

H = 400/(30+X)

We do not know whether X is longer or shorter than 30. If X is longer, then the slant length 18 is the hypotenuse of a right triangle with sides 400/(30+X) and (X-30)/2. If X is shorter, then 18 is the hypotenuse of a right triangle with sides 400/(30+X) and (30-X)/2. See diagram below.

Whether X is longer or shorter than 30, the Pythagorean Theorem gives the equivalent equations

[400/(30+X)]^2 + [(30-X)/2]^2 = 18^2

[400/(30+X)]^2 + [(X-30)/2]^2 = 18^2

Expanding the squared terms and clearing the denominators gives the quartic polynomial equation

X^4 - 3096X^2 - 77760X + 283600 = 0

The real-valued solutions to this equation are X ≈ 3.232497 and X ≈ 65.001253. Both of these produce trapezoids with slant lengths of 18 and areas of 200. Therefore, Case II yields two solutions {30, 3.232497, 18, 18} and {30, 65.001253, 18, 18}.

**Case III:** If we assume one of the parallel sides is 18 and the slanted sides are 30, following the method of Case II above, the missing parallel side X can be found by solving the equivalent equations

[400/(18+X)]^2 + [(18-X)/2]^2 = 30^2

[400/(18+X)]^2 + [(X-18)/2]^2 = 30^2

Simplifying produces the quartic equation

X^4 - 4248X^2 - 129600X - 421424 = 0

The real-valued solutions to this equation are X ≈ -3.698696 and X ≈ 77.411244. Only the positive value of X can be considered a solution. Therefore, Case III yields the solution {18, 77.411244, 30, 30} as the sides of the trapezoid.

**Summary:** As you can see from this example, if you know two of the sides of a symmetric trapezoid and the trapezoid's area, but you don't know which sides are parallel and slanted, then you can obtain multiple correct solutions.

## Example 2: Asymmetric Trapezoid, 4 Solutions

An asymmetric trapezoid has sides of length 13, 13, 34, and an area of 282. What is the missing side length?

In this problem, although we are not told whether the missing side is a parallel side or a slant side, the fact that the trapezoid is asymmetric tells us that the two sides of length 13 cannot both be the slant sides since would make the trapezoid symmetric. Moreover, the two parallel sides cannot both have a length of 13, since this would make the shape a rectangle or parallelogram. This leaves only two cases to consider.

- Case I: 34 and 13 are parallel sides, 13 is slant side, missing side is slant
- Case II: 34 and 13 are slant sides, 13 is a parallel side, missing side is parallel

These two cases are shown in the diagram below.

**Case I:** This is problem where we can use the 4-side area formula for the volume of a trapezoid. If a trapezoid's parallel sides are P and Q and the slant sides are R and S, then the area is given by the formula

Area =

[(P+Q)/(4P-4Q)]*sqrt{ 2[(RS)^2 + R^2(P-Q)^2 + S^2(P-Q)^2] - [R^4 + S^4 + (P-Q)^4] }

If we have P = 34, Q = 13, S = 13, and Area = 282, then we can solve for the other slant side R. This gives us the equation

282 = (47/84)*sqrt[ -R^4 + 1220R^2 - 73984 ]

Solving for R gives us

504^2 = -R^4 + 1220R^2 - 73984

R^4 - 1220R^2 + 328000 = 0

(R^2 - 400)(R^2 - 820) = 0

R = 20, -20, sqrt(820), -sqrt(820)

Only the positive values of R yield solutions for the missing trapezoid side. Therefore, Case I yields the two solutions { 34, 13, 13, 20} and {34, 13, 13, 28.63564}.

**Case II:** In the second case, we can also solve for the missing side using the 4-side area formula. Here we have Q = 13, R = 34, S = 13, Area = 282, and P is the unknown side length. Plugging these values into the formula we have

282 =

[(P-13)/(4P-52)]*sqrt{ 2[195364 + 1325(P-13)^2] - [1364897 + (P-13)^4] }

This reduces to another quartic equation whose real-valued roots are P ≈ 38.249444 and P ≈ 56.127504. Therefore, Case II yields the solution sets {34, 13, 13, 38.249444} and {34, 13, 13, 56.127504}.

**Summary:** Once again, we see that a missing side problem can yield multiple correct solutions if the initial question comes with few constraints.

## Example 3: Symmetric Trapezoid, 1 Solution

Here is an easier problem in finding the missing side length of a trapezoid with only one correct solution. Suppose a symmetric trapezoid has side lengths of {24, 20, 10} and an area of approximately 216. What must the missing side length be?

Since the trapezoid is symmetric, two of the sides (the slanted sides) must have equal length. This means the possible solutions are

- Case I: sides = {24, 24, 20, 10}
- Case II: sides = {24, 20, 20, 10}
- Case III: sides = {24, 20, 10, 10}

To determine which one is the correct solution, we can use the Pythagorean Theorem to find the height of the trapezoids in each case, compute the areas using the base-height formula, and see which side lengths produce a trapezoid with an area of approximately 216.

**Case I:** The slant side of length 24 is the hypotenuse of a right triangle with a leg length of 5. The other leg is the height of the trapezoid, equal to sqrt(24^2 - 5^2) ≈ 23.4734. Using the base-height trapezoid area formula, the area of this shape is 23.4734*(10+20)/2 = 352.101. This area is not close to 216, therefore this is not the solution.

**Case II:** The slant side of length 20 is the hypotenuse of a right triangle with a leg length of 7. As before, the other leg is the height. The Pythagorean Theorem gives us height = sqrt(20^2 - 7^2) ≈ 18.735. The area of the figure is 18.735*(10+24)/2 = 318.495. Again, this answer is too large, so Case II does not provide the solution either.

**Case III:** The slant side of length 10 is the hypotenuse of a right triangle with a leg length of 2. This means the height equals sqrt(10^2 - 2^2) ≈ 9.79796. The area of the trapezoid is 9.79796*(20+24)/2 = 215.55512, which is 216 when rounded up to the nearest integer. Therefore, the sides of the trapezoid are {24, 20, 10, 10}

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