PiRSquared: Simple Area Formulas: Rectangle, Parallelogram, Triangle, Trapezium

I'm aware that most people have already covered/learned these formuals. However, they're incredibly important no matter what level of math you're at, they're easy to learn, and they make a good first math-y post. :)

The formulas, without explanation, are as follows:

Where "A" equals the area of a shape, b equals the length of the base, and h equals the height of the highest side in relation to the bottom edge.

Rectangle: A = b x h

Parallelogram: A = b x h

Triangle: A = ½ x b x h

Trapezium: A = ½ x h (x + y)

In the last formula, for the trapezium, x and y are the parallel sides.

If you just need to figure out the area right away, those are good to use.

But if you're like me, and you need to understand exactly why something works- or at least the general idea- then here are my explanations.

Rectangle: A = b x h

(Area equals base multiplied by height)

Why?

Take a look at my illustration.

Judging from my picture, if you take A = b x h, that's A = 4 x 3. Anyone who has learned simple multiplication will know that that means that the area of this rectangle is 12².

Again, this is an incredibly simple explanation. To understand, simply draw the lines along the measurement markers...

Count the squares (or rectangles, considering my lousy job with my illustrations) and you'll see that there really are twelve squares, leading to yet another way to demonstrate this simple formula.

Parallelogram: A = b x h

(Area equals base multiplied by height)

Why?

And I gave up on trying to make them
look neat... X_X

You may notice that the formula for finding this shape is the exact same as it was for finding the area of the rectangle. The reason for this (which you can possibly see from my sloppy illustration) is because a parallelogram is, essentially, a squashed rectangle. But wait, you might be thinking. That can't be right- after all, it looks more like a square with two triangles shoved on!

Yes, it is, in a way. Here's another illutration showing how we can make it look more like a simple rectangle... By taking triangle YZX and moving it to the other side, we form a perfect rectangle. Divide that along the measurement marks (just like we did with the rectangle) and you can see how this formula works. (Again, the area of this parallelogram is 12², just like it was for the rectangle.)

Triangle: ½ x b x h

(One half multiplied by the base multiplied by the height)

Why?

The reason for this formula is quite simple. Take triangle ABC. Its base is 3 and it has a height of 4. If you add another triangle of the same dimensions (or, considering this is an iscoceles triangle, split the congruent triangle A'B'C' in half) and rearrange it, you get this rectangle. Your original triangle is half of the said rectangle (or square, if your original triangle is an equalatreral) leading to A = b x h (area of your rhombus) divided by half, leading to the area of your triangle.

Tip: When solving it, it's best to align your problem above the 2- especially if you have decimals.

Trapezium: A = ½ x h (x + y)
(Area equals one half multiplied by height multipled times the sum of the two parallel sides)

Why?

First let's take a look at the shape itself.

A trapezium has two parallel lines, with one being shorter than the other. The other two sides are diagonal- although the angles against x and y are not always equal.

When you first look at the formula for finding the area of a trapezium, it can look a little bewildering. But the reasons why you use this formula- instead of finding the area of the rectangle/square in the center, then the area of the two triangles on the sides, then adding them together- really does make sense. The next illustration shows basically what we're doing to this poor trapezium when we use this formula.

Basically, we're taking the triangle from one side of the trapezium, cutting it off, flipping it upside down, and then applying it to the other side of the original shape, which creates a rectangle or a square. We can do this because the two sides (x and y) are parallel, meaning that they're the same difference away from each other on both sides, meaning that the places where the triangles are attached (first trapezium illustration- the dotted line is this place where the triangle is attached) are the same length on both sides.

To be honest, the ½ is applied to the (x + y) section of the formula. Take another lok at our example trapezium. x + y = 16. 16 x ½ (or 16 divided by two) = 8. Go back up and look at our transformed trapezium. The two parallel sides now equal 8- we have equalized their lengths. Because the distances between the two are constant, we're left with a rectangle.

Because I have time, and because 7, 8, and 9 have so little differences between the numbers, let me show you how this formula works for a trapezium with a greater number difference...