Half the sum of the parallel sides, times the height between them, that is how you calculate the area of a trapezium.

And that is what we learnt at school (to the tune of pop goes the weasel.)

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Here's some more detail, for those who want it:

For a trapezoid where h is the height (ie: the distance between the two parallel sides), and a & b are the lengths of the two parallel sides:

Area = h (a + b) / 2

Proof:

If you look at a trapezoid sitting on the longer of its two parallel sides, you will see that it's actually a rectangle in the middle, with a right-angled triangle on each side.

The area of the rectangle would be:

h x a (where a is the shorter parallel side)

For the two right-angled triangles, if c and d are the lengths of the perpendicular sides, then the areas are:

(h * c) / 2 and

(h * d) / 2

So, we add the two triangle areas together, and cancel:

(h * c + h * d) / 2

= h (c + d) / 2

This can be substituted for the following, since in a trapezoid a + c + d = b, hence c + d = b - a

h (b - a) /2

So, the total area of the trapezoid is the sum of the two formulas:

h (b - a) / 2 + h * a

(multiply the "h * a" part by 2, and divide be 2 as well, to get a common denominator):

= h (b - a) / 2 + 2ha / 2

= h (b - a + 2a) / 2

= h (b + a) / 2

So, there's your proof. A little complicated, but hopefully you get the idea.

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