Splitting a Rectangle to Find Its Area

Area counts squares

The area of a rectangle is the number of unit squares that fill it.

A rectangle $4$ tall and $7$ wide holds $4$ rows of $7$ squares:

$$4 \times 7 = 28 \text{ squares}$$

That is a hard fact for many people. There is a way around it.

Cutting the rectangle

Draw a line down the rectangle. Now there are two smaller rectangles.

The left part is $4$ tall and $5$ wide. The right part is $4$ tall and $2$ wide.

Part	Squares
left	$4 \times 5 = 20$
right	$4 \times 2 = 8$
both	$20 + 8 = 28$

That matches the $28$ from before.

Why the total cannot change

The line is drawn on the squares. It does not move them.

Every square that was inside the big rectangle is still inside one of the two parts. No square was added. No square was lost.

So the two counts have to add up to the first count.

$$4 \times 7 = (4 \times 5) + (4 \times 2)$$

Why this makes multiplying simpler

The $7$ was the hard part. Splitting it into $5$ and $2$ leaves two facts most people know.

$$4 \times 5 = 20 \qquad 4 \times 2 = 8 \qquad 20 + 8 = 28$$

A fact you do not know becomes two facts you do. The picture is what makes it safe: you can see that nothing was left out.

You choose where to cut. Pick the split that gives you the easier pair.

To find	Split as	Two facts
$6 \times 7$	$6 \times 5$ and $6 \times 2$	$30 + 12 = 42$
$8 \times 9$	$8 \times 10$ take away $8 \times 1$	$80 - 8 = 72$
$3 \times 14$	$3 \times 10$ and $3 \times 4$	$30 + 12 = 42$

The last row is how bigger numbers are handled. Splitting $14$ into $10$ and $4$ is exactly what column multiplication does.

The rule behind the picture

This rule has a name. It is the distributive property.

$$a \times (b + c) = (a \times b) + (a \times c)$$

The letters stand for any numbers you like. The rectangle is the reason it is true: $a$ is the height, and the width $b + c$ is cut into $b$ and $c$.

You will meet it again with letters instead of numbers in algebra. It is the same picture.

Worked examples

Common mistakes

Practice problems