How do you find the area of a trapezoid when you have the length of every side but not the height?

1 Answer
Nov 26, 2015

Let's say that you have the length of every side and they are named #a#, #b#, #c# and #d# like in my drawing, with #b# and #d# being base sides and #d# being the longer base.

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The formula to compute the area of trapezoid is:

#A = (b+d)/2 * h#,

so basically the only thing we need to do is computing #h#.

If you draw the height #h# like in my drawing, you see that the long base #d# can be divided into three intervals: #d = d_1 + d_2 + d_3# with #d_2 = b# and #d_1# and #d_3# yet unknown.

However, we also have two right angle triangles, one with the legs #d_1# and #h# and the hypotenuse #a# and one with the legs #d_3# and #h# and the hypotenuse #c#.

So, we know three things (with the known values marked in #color(green)(green)#):

  1. #color(white)(xxx) d_1 + d_3 = color(green)(d) - color(green)(b)#

  2. #color(white)(xxx) d_1 ^2 + h^2 = color(green)(a^2)# (theorem of Pythagoras)

  3. #color(white)(xxx) d_3^2 + h^2 = color(green)(c^2)# (theorem of Pythagoras)

Now, you can solve the equation in 1. for e.g. #d_1# and plug it into equation 2. or 3.

Afterwards, the equations can be solved, and you will get the values for #d_1#, #d_3# and #h#.

And as soon as you have #h#, you can appy the formula and compute the area!

Please also check here an example where I've done exactly the same calculation on concrete values. :-)

http://socratic.org/questions/what-is-the-area-of-a-trapezoid-with-base-lengths-of-12-and-40-and-side-lengths-#193240