Solve one side of irregular quadrilateral with known area (formula help)

Question

My question is about determining the formula for a missing length of a known area irregular quadrilateral.

Its a fairly easy one, but i'm about 40 years outside of my math lessons!

It's a land area problem - and I know the area, but don't know the frontage. I know each side. So the shape is a triangle on top of a square.

Example: Left side = (LS) = 18M

Right side = (RS) = 14M

Area = 400 sq M

Unknown is the length of the square (B)

I know the area of the triangle + area of the square = 400,

which is: 400 = (1/2 * B * (LS-RS)) + (B * RS)

But I can't distill that down to an equation! I know the answer in this case is B=25, but I had to test numbers to find it.

I'd like to solve for an equation for different values.

Alternatively, if there's an easier way to know the area, I'm very happy to learn!

Answer 1

So you are working towards rediscovering the formula for the area of a Trapezium. This is a quadrilateral with a pair of parallel sides - let's call them left and right, and let's call the perpendicular distance between those sides the base. Then the formula you have for the area is: $$A=\frac b2(l-r)+br$$and if we work just a little with this we get $$A=\frac {bl-br+2br}{2}=b\left(\frac {l+r}2\right)$$

The area is the base times the average length of the parallel sides. (The area is the average length of the parallel sides times the perpendicular distance between them)

In your case the average length of the parallel sides is $16$ and $400/16=25$

Note that the formula works for a rectangle, where the parallel sides are equal and you just get base times height, and for a triangle, where taking one side as being of zero length you get half base times height. But that is no surprise, because those are the components which made up the calculation. It also works for a rectangle with a triangle added at each end (provided the figure you get is a quadrilateral with a pair of parallel sides).