The area of a flat shape is the size of the space inside the shape.
For the area, yes, there will be a few rules.
I’m going to try to make it simpler for you by showing you the connections between
some of the shapes, and where some of the rules come from.
Let’s start with a rectangle.
The area is length times width.
Most people know that.
But just in case it isn’t obvious to you, let me explain.
Imagine I break up the sides of my rectangle into little even pieces.
Then the area is just going to be the number of squares I have in the middle of my rectangle.
In fact, that’s a perfectly good way to estimate the area of any shape, even an irregular
one with curvy sides and an unusual shape.
Just draw a neat, square grid over the top of it, and count the squares.
And of course a square is just a special kind of rectangle where the length is the same
as the width, so the area of a square will be length times length, or side length squared.
Some people learn this as a separate formula, but there’s really no need.
It’s just a special case of the rule for any rectangle.
What about a parallelogram?
Well, would you believe the rule is just length times width again ? The key is, the width
must be perpendicular to the length, the perpendicular width of the shape.
(Perpendicular means “at right angles”.)
Not the length of the sloping side.
You need the perpendicular width.
You need two measurements at right angles.
When you have a length and a perpendicular width, you just multiply them to get the area.
Let me show you something that might help to explain this.
Look at this parallelogram.
If I cut this piece off here, and move it over to the other side, what shape have I
got now?
And what was the rule for the area of a rectangle?
Length times width.
See?
As long as you use the actual width of the shape, the width of the parallelogram and
not the length of the sloping side, you’re basically just working out a rectangle.
And of course if you can work out a parallelogram you can also do a rhombus exactly the same
way, because a rhombus is just a special kind of parallelogram, where the sides are all
equal.
What about a triangle?
Well, some of you may remember the rule as half base times height.
But it’s actually just another special case of the same rule we’ve just been learning.
It’s actually half of width times length.
Let me show you.
Look at this triangle.
If I draw new sides up like this, what have I got now?
It’s a rectangle.
And this new part of the rectangle is the same area as this half of the original triangle.
And this new bit over here is the same area as the other half of the original triangle.
So that means that the triangle I started with is exactly half of the area of the rectangle
around it.
What’s the length of the rectangle?
And what’s the width?
As long as you have two measurements for the triangle that are at right angles, two perpendicular
measurements, you can plug them in, and you’ll get the area.
Just don’t forget to take half of it for a triangle.
But hang on, what about triangles like this one?
Where do I measure the width?
What’s the perpendicular height of this triangle?
Well, where’s the base?
Which side am I going to call the length?
Now, what’s the height?
What’s the width, perpendicular to the length?
So, just use that.
Not convinced?
What if I draw in another identical triangle here?
Now what shape have I got?
And the original triangle is exactly half of the area of the parallelogram.
Half length times width.
To be honest, I usually just use “half base times height” myself, because that’s the
way I learnt it.
But I do think this method makes it much clearer why it works.
It also helps explain why this next formula works.
This shape is a trapezium.
In the formula, a and b are the parallel sides, and h is (again) the perpendicular height.
a and b are the lengths of the parallel sides, and h, the height, is the width of the shape
perpendicular to the lengths.
If you can’t remember the fancy formula, or if you get stuck trying to remember which
side is a, b or h, you can always just break it up into smaller pieces.
You could do it like this, with one rectangle and two triangles.
Or, watch what happens if instead I just make it two triangles, like this.
The area of this one is half its base, a, times its height, h.
And the area of this one is half its base, b, times the height, h.
Hmm, that’s interesting.
If I factorise that using the distributive law, what do I get?
And finally, you really will just have to learn the rule for the area of a circle.
It’s pi r squared.
Please, please, please be careful with the way you say this.
“Pi r squared”, not “pi r two”.
It’s not r-times-two, it’s r-times-r, r squared.
Get the language right, and your understanding will follow.
Get it wrong, and it’s just too easy to make a silly mistake.