Unit 15: Area

Area counts squares

Area is how many unit squares fit inside a shape.

A rectangle 5 across and 3 up holds 3 rows of 5 squares, so its area is 5 × 3 = 15. That is the only area fact anyone needs to be told — everything else is built from it.

Area is measured in square units: cm², m², and so on.

A triangle is half a rectangle

Draw a rectangle round any right-angled triangle and the triangle is exactly half of it.

For a triangle that is not right-angled, drop a straight-down height and it splits into two right-angled ones — and each of those is half its own rectangle.

Either way: area = base × height ÷ 2.

Cut it up, or fill it in

An awkward shape has two standard attacks.

Cut it up into rectangles and triangles, work out each, and add.

Or fill it in to make a big rectangle, work that out, and subtract what you added.

Whichever needs fewer pieces is the right one. Both give the same answer, which is a good way to check yourself.

A hundred, not ten

There are 100 cm in a metre. So how many cm² in a m²?

Ten thousand. A square metre is 100 cm across and 100 cm up, so it holds 100 × 100 little squares.

Length converts by the factor. Area converts by the factor squared. This catches out more people than any other conversion.

Choosing sensible units

A postage stamp is measured in cm². A room in m². A farm in hectares — a square 100 m on each side, so 10000 m².

A country goes in km², and one km² is a million m².

Picking the right unit keeps the numbers readable, which is the only reason so many units exist.

One measures the fence, the other the grass

Perimeter is the distance round the edge. Area is the space inside. They are not related in any simple way.

Take two rectangles that both have perimeter 20. One is 1 by 9, with an area of 9. The other is 5 by 5, with an area of 25.

Same length of fence, nearly three times as much grass. Knowing one of these tells you very little about the other.

The square always wins

Of all rectangles with a fixed perimeter, the square holds the most area.

And of all shapes at all, the circle beats even the square — which is why bubbles are round and why animals curl up to keep warm.

A farmer with a fixed length of fence should think hard about the shape before hammering in a single post.

Box it in and cut away

A triangle at an awkward angle on a grid has no obvious base or height. So do not look for one.

Draw the smallest rectangle that contains it. Work out that area, then subtract the right-angled triangles in the corners — each of which is half a rectangle and easy.

This works for any shape whose corners sit on grid points.

Counting squares

For a wobbly shape, counting works. Count every whole square inside, then pair up the part-squares — two halves make a whole.

It is not exact for a curve, but it gives a good estimate, and it is how people measured lakes on maps for centuries.

And it makes the point that area really is a count of squares, not a formula.

Moving a piece changes nothing

Cut a shape up and rearrange the pieces. The area is exactly the same — you have not added or removed anything.

That is the single idea behind every area formula in this unit. The parallelogram became a rectangle. The circle became a rectangle. The triangle is half of one.

So when a shape looks impossible, ask what it could be cut into.

The same area, two different ways

A good check: work an area out by two different routes and see whether they agree.

An L shape can be seen as two rectangles added, or as one big rectangle with a bite taken out. Both must give the same number.

When they do not, you have found a mistake — which is a much better outcome than not knowing.

Lengths once, areas twice

Make a shape 3 times as long in every direction. What happens to the area?

It goes up by 3 × 3 = 9. The width grew and the height grew, and area is the two multiplied.

So a scale factor of k on the lengths is a scale factor of k^2 on the area. This catches people out for their entire lives.

Same height, sliding along

A triangle keeps the same area if you slide its top point sideways, as long as the height does not change.

So two triangles with the same base and between the same pair of parallel lines are equal in area — however different they look.

This is the tool that cracks "find the shaded area" puzzles.