Calculating Trapezoid Area


This is a method for calculating the area of a trapezoid, without using the trapezoid height or area formulas.
Let's use this trapezoid as an example.


First, we need to calculate the trapezoid height.
To do this, we drop perpendiculars (called 'hgt' for height) from the upper base to the lower base, producing the image you see below:

Using the Pythagoren Theorem, we can create two equations for the trapezoid height.

hgt2 = 152 -L2   or   hgt2 = 225 -L2
hgt2 = 202 -R2   or   hgt2 = 400 -R2

Since the right side of both equations = hgt2, we can state that both equations equal each other:
225 -L2 = 400 -R2   or
a) L2 = R2 -175

We know that L + R + 30 = 55 so, we can create 2 more equations:
L = 25 -R   and   R = 25 -L
If we square L = 25 -R we get the equation:
b) L2 = R2 -50R +625

Since the right side of equations 'a' and 'b' equal L2, we can state that:
R2 -175 = R2 -50R +625
50R = 800
R = 16

Now that we have a value of R, we can use the Pythagorean Theorem to calculate the height:
hgt2 = 202 -162
Trapezoid height = 12

We know from another equation that L = 25 -R
L = 25 -16
L=9

Now, we can calculate the entire area:

Triangle Areas = (½*9*12) + (½*16*12) = 54 + 96 = 150
Plus
Rectangle Area = 30 * 12 = 360

Trapezoid Area = 510

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