The Murderous Maths Research Lab proudly presents...

THE TRAPEZIUM AREA PAGE

This is possibly the most exciting page on the whole internet. Or the dullest. It depends on you.

A trapezium is a shape with four sides, two of which are parallel. The sides can all be different lengths. (Note: in the US this shape is called a TRAPEZOID)

Here's one that we photographed earlier. You'll notice that the sides with lengths a and c are parallel.

Usually if you want to find the area of a trapezium you use this formula:

Area of a trapezium = h(a+c)/2

Trapeziums... If you have a parallelogram (where both pairs of opposite sides are parallel) or an irregular quadrilateral, then you can't work out the area by just knowing the lengths of the sides! But so long as you know which two sides are parallel, you can work out the area of a trapezium.

The U.S. TRAPEZOID The US defines a trapezium as having NO parallel sides, and a quadrilateral with just 2 parallel sides is called a TRAPEZOID. Let's all sing together now: "You say trapezoid, and we say trapezium... let's call the whole thing off."

In the formula, a and c are the lengths of the two parallel sides, and "h" is the height of the trapezium. So far so simple BUT...

...we were asked if we could work out the area if we knew the lengths of the four sides, but we DIDN'T know the height! Obviously the usual formula wasn't going to help much, so with the help of some of our regular visitors to this site, we came up with another formula.

The first sensible suggestion came from our Singapore friend Hu Yi Jie. He suggested splitting up the trapezium up into three bits and then attacking the problem like this:

• You chop off the two triangles at the end to leave a rectangle in the middle. You then put the two triangles together to make one big triangle. We know the lengths of all three sides of this triangle.
  The bottom of the triangle measures (a-c) because the bottom of the trapezium was a, and we have just removed a section that measures c. The other two sides measure b and d.

• If you've read The Perfect Sausage you'll know about semi-perimeter cows!

  You can work out the area of the triangle using the "semiperimeter" formula.
  This brilliant formula lets you work out the area of a triangle if you know the lengths of the three sides! The "semiperimeter" is half the distance around the triangle so if the lengths of the triangle sides are X,Y and Z then s=½(X+Y+Z). Once we have "s", we get the triangle area with the formula:
          TRIANGLE AREA= (s(s-X)(s-Y)(s-Z))^½
  We know that for OUR triangle the side lengths are b,d and (a-c) so first you work out the semiperimeter s=½(b+d+a-c) and then plonk everything into the area formula to get:
  TRIANGLE AREA = (s(s-b)(s-d)(s-(a-c)))^½. Nice and easy isn't it?
• The most common formula for the area of a triangle is AREA = base x height/2. As we now know the area and we already know the base=(a-c) we can work out the height "h" of the triangle.
  So h = 2(s(s-b)(s-d)(s-a+c))^½/(a-c)
• And here's the good bit... the height of the triangle is also the height of the rectangle, so you can then work out the area of the rectangle.
  The rectangle area is c x h therefore the RECTANGLE AREA = 2c(s(s-b)(s-d)(s-a+c))^½/(a-c)
• Add together the areas of the triangle and rectangle... and that's it!

AREA OF TRAPEZIUM = (1 + 2c/(a-c)){s(s-b)(s-d)(s-a+c)}^½

where s=½(b+d+a-c)

And when you swap every "s" in the main formula with ½(b+d+a-c) and boil it down you get our formula for the area of a trapezium...

Of course, we've only shown you Hu Yi Jie's EASY version here. We had already worked out another way of solving the problem which we think is even easier. But then we would wouldn't we?

• First we extended the two non-parallel sides of the trapezium so that they touched. This made two similar triangles.
• There's enough information to work out the lengths x and y in terms of a,b,c and d.
• Using the semiperimeter formula again, we worked out the areas of the two triangles and then took the small one away from the big one to leave the area of the trapezium.

Amazingly enough we found that Hu Yi Jie's method produced exactly the same formula as ours did... AND IT WORKS!

We tested it on a trapezium with sides a=30, b=10, c=9 and d=17. This conveniently makes a trapezium of height 8, and it breaks down into a rectangle measuring 9x8 (area=72), and two right angled triangles of 6-8-10 (area=24) and 8-15-17 (area=60). The trapezium area is 72+24+60=156 which is what the formula gives. Oooooh.... we're so pleased with ourselves.

TOUGHER FORMULAS

A special big cheer to Jenny Wood, and also Carl "Maths Whizz" Turner who both came up with answers using the cosine and sin rules to work out the angles of the trapezium, and then the height and then the area. We haven't tested these formulas, and at the moment we haven't time to explain the COS and SIN rules here. However, if you're a super brainy person we've typed them out as neatly as we can just for YOU and here they are ....

JENNY'S AREA OF TRAPEZIUM = ½(a+c)*b*sin{cos^-1[(d²-(a-c)²-b²)/(-2(a-c)b]}

CARL'S AREA OF TRAPEZIUM = (a+c)(½bd(sin(cos^-1(b²+d²-(a-c)²))))/4(a-c)

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