# Trapezoid Area Given 4 Sides

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

Usually, the area of a trapezoid or trapezium is given as a function of the height and lengths of two parallel sides. But what if you don't know the height? Instead, what if you only know the lengths of all four sides? It turns out this is enough information to figure the area or volume of a trapezoid shape so long as you know which two lengths correspond tot he parallel sides and which two lengths correspond to the slanted sides.

This tutorial will work out several examples of finding trapezoid area from four side lengths. It also includes an example of what can happen when you know the four sides but don't know which two are parallel.

## Trapezoid Area Formula with 4 Sides

If the two parallel sides of the trapezoid have lengths P and Q with P being the longer side, and the two slated sides have length R and S, then the area of the trapezoid is given by the formula

Area =
[(P+Q)/(4P-4Q)] * sqrt{ 2[R²S² + R²(P-Q)² + S²(P-Q)²] - [R⁴ + S⁴ + (P-Q)⁴] }

Though it looks more complicated than the usual formula H*(P+Q)/2, it is much easier to work out than trying to find an unknown height. If this trapezoial area equation looks similar to Heron's formula for the area of a triangle, it's no coincidence as we will see in the next section.

## How the Formula Is Derived

Every trapezoid can be thought of as the difference between two similar triangles, and the area of a trapezoid is the difference in area between those triangles.

The slanted sides of a given trapezoid can be extended so that they meet at a point, forming a triangle. If the sides of the trapezoid are known, then the sides of the larger and smaller triangles can be figured using proportions. These side lengths are given in the following diagram.

Extending the sides of a trapezium with sides {P, Q, R, S} you get a small triangle with sides {Q, QR/(P-Q), QS/(P-Q)} and a larger similar triangle with sides {P, PR/(P-Q), PS/(P-Q)}. The area of the larger triangle minus the area of the smaller triangle is the area of the trapezoid.

Heron's formula gives the area of a triangle in terms of its three side lengths, if a triangle's sides are {x, y, z} then

H(x, y, z) = 0.25*sqrt{ 2(x²y² + x²z² + y²z²) - (x⁴ + y⁴ + z⁴) }

The area of the trapezoid is thus

H( P, PR/(P-Q), PS/(P-Q) ) - H( Q, QR/(P-Q), QS/(P-Q) )

Since H(ax, ay, az) = (a²)H(x, y, z) for any constant "a", we can greatly simplify the expression to obtain the area of the trapezoid as [(P²-Q²)/(P-Q)²]*H(P-Q, R, S) = [(P+Q)/(P-Q)]*H(P-Q, R, S), which matches the form of the area equation given above.

## Example 1

A trapezoidal plot of land has sides of length 60 meters, 32 meters, 40 meters, and 43 meters. The sides of length 60 and 32 are parallel, while the sides of length 40 and 43 are slanted. Using the values P = 60, Q = 32, R = 40, S = 43, and P-Q = 28 we get

Trapezoid Area =
[(P+Q)/(4P-4Q)]*sqrt{ 2[R²S² + R²(P-Q)² + S²(P-Q)²] - [R⁴ + S⁴ + (P-Q)⁴] }
= [92/112]*sqrt{ 2[5662416] - [6593457] }
= [23/28]*sqrt{4731375}
≈ 1786.75 square meters
0.4415 acres

## Example 2

A trapezium has sides 4, 5, 6, and 9. The side of length 9 is one of the parallel sides, but it is not known how long the other parellel side is. What could be its area?

In this problem, there are 3 possible configurations for the trapezoid, pictured below.

Case I: If the other parallel side is 4, then we have P = 9, Q = 4, R = 5, and S = 6. Plugging these four values into the area formula gives us an area of exactly 156/5 = 31.2

Case II: If the other parallel side is 5, then we have P = 9, Q = 5, R = 4, and S = 6. Plugging these four values into the area formula gives an exact area of (7/8)sqrt(1008), or approximately 27.7804. This figure might not look like a traditional trapezoid because the slant sides lean in the same direction, but it does fit the technical definition of trapezoid.

Case III: If the other parallel side is 6, then we have P = 9, Q = 6, R = 4, and S = 5. Plugging these four numbers into the formula gives an exact area of 30. This is an example of a trapezoid with one slant side and one perpendicular side.

The area of the trapezoid may be either 27.7804, 30, or 31.5.

## Example 3

A trapezoidal prism has a volume of 13,524 cubic inches. What is known about its dimensions are that the length of the prism is 49 inches, the two parallel sides of the trapezoidal cross-section have lengths 30 and 16 inches, and one of the slant sides of the trapezoidal cross-section has a length of 15 inches. What is the length of the other slant edge?

The volume of a trapezoid prism is the area of the trapezoid cross-section times the length of the prism. This gives us

49*Area = 13524
Area = 13524/49
Area = 276

Letting P = 30, Q = 16, and R = 15, we need to solve for S. The formula for the area of a trapezoid given four sides tells us that

276 =
(46/56)sqrt{ 2[225*S^2 + 196*S^2 + 44100] - [50625 + S^4 + 38416] }

336 = sqrt{ -S^4 + 842*S^2 - 841 }

112896 = -S^4 + 842*S^2 - 841

S^4 - 842*S^2 + 113737 = 0

(S^2 - 169)(S^2 - 673) = 0

S = 13, -13, sqrt(673), -sqrt(673)

The two positive solutions for the value of S are 13 inches and sqrt(673) ≈ 25.94224 inches. These trapezoids look like this:

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