## Areas of Simple Shapes

The formulas to work out the areas of most shapes are fairly simple.  However, the best way to learn them is to understand how the formulas were created in the first place.  By doing this, even if you forget the exact formula in an exam you can usually work it out for yourself!  So here are some brief explanations about how each formula was created for each shape.

### Area of a triangle

If you draw a triangle like the shaded area in the diagram you can always draw a rectangle around it like in the diagram.  Look how the triangle (the shaded area) can be split into two separate bits.  Each of those bits has a matching area in the rectangle.  The matching areas are shown by the double headed arrows in the diagram.

We already know that the total area of a rectangle is just the length multiplied by the width.  In this diagram we’ve got different symbols to L and W – instead we’ve got ‘b’ and ‘h’.  So the area of our rectangle is just ‘b’ multiplied by ‘h’ – ‘bh’.  Now what fraction of this area is taken up by the triangle?  Well, the triangle takes up exactly half of the area of the rectangle.  This is because each bit of the triangle has another matching bit in the rectangle.  If you wanted to you could take the part of the rectangle which isn’t in the triangle and make another triangle like this:

This diagram shows how you can rotate the remaining bits of the rectangle around to make another identical triangle.  This demonstrates that two of these triangles fit inside the rectangle.  This means that the area of each triangle is half of the rectangle’s area.  So if the rectangle’s area is ‘bh’, then the triangle’s area is .  Be careful to use the vertical height in your calculations, like it’s labelled in the first diagram.

### Area of a parallelogram

It’s easy to cut a bit off a parallelogram and move it to the other side so that you get a rectangle.  We all know the area of a rectangle is just its length multiplied by its height or width.  So the area of a parallelogram is just its vertical height multiplied by the length of one of the parallel sides.

### Area of a trapezium

A trapezium has four sides, with two of the sides parallel.  Like for a triangle and parallelogram, you can calculate its area by putting a rectangle over it and re-arranging bits like this (you might need to rotate the bits to make them fit):

So what you get by re-arranging these two bits is a rectangular area.  The height of the rectangle is the same as the vertical height of the trapezium, but how long is its other side?  Well, it’s longer than ‘a’, but shorter than ‘b’.  In fact, it’s exactly halfway between ‘a’ and ‘b’ in length – so it’s the average of ‘a’ and ‘b’.  The average of ‘a’ and ‘b’ is half of them added together: .  So the area of a trapezium is just this length times the vertical height:

### Area of a circle

There are a few ways to work out the area of a circle.  Like for most of the other shapes, it involves cutting up the shape and re-arranging it.  In this case, we need to cut the circle up into lots of little wedges, just like you were cutting a cake.  Then we can pull out these wedges one by one and line them up like in the diagram.  Now, we know that the circumference or perimeter of the circle is , this is shown in the diagram around the circle.  When we line up the wedges, the length of the side made up of all the wedge ends is also going to be the same length - .  The height of this row of lined up wedges is going to be the radius of the circle – R. The  symbol is the lowercase version of the capital symbol for pi, which is .

Now what we can do is split this line of wedges up into two parts, and then flip one vertically and push it against the other one.  This will halve the length of the rows to just :

When they get pushed together, you just get a rectangle:

Now we all know that the area of a rectangle is its length multiplied by its height.  So in this case, it’s:

And that’s how you get the formula for the area of a circle.

### Area of a sector

A sector is a pie shaped wedge of a circle.  To work out what its area is, we just need to find out what fraction of the circle the sector is occupying.  Now we’ve already worked out that a sector with a sector angle of 360° is the same as a circle.  This means it would have the same area as the whole circle.  What about a sector angle of 180°?  Well this would only be half a circle, so it would only have half the area of a circle.

So the fraction of the circle area that the sector takes up is just the sector angle divided by the angle you’d need to get a sector like a circle, which is 360°.  In mathematical terms, the formula for the area of a sector is:

Now we know that the area of a circle is , and we’ve just worked out that the fraction of the circle the sector occupies is the sector angle divided by 360°.  So we can rewrite the formula:

Here is a summary of some area formulae:

 Area = L2 Area = L × W Area = Area = Area = bh Area =

### Things to watch out for

There are only a few tricky things to look out for when you’re doing simple area questions.  First of all, sometimes questions will mix units – so for instance they might have centimetres and metres in the same question.  Make sure you work using only one type of unit – so you may have to convert one of the units into another type.  For instance, if you chose to work in centimetres, you’d have to convert all metre lengths into centimetres by multiplying by 100.

Second thing you have to watch out for is problems which have a combination of shapes in them – a question may have both rectangles and circles in them.  Also, sometimes you’ll only get parts of shapes, rather than the whole shape.