Area of a rectangle

Counting unit squares, and why base times height counts them for free.

Area is a count: how many unit squares fit inside a shape. For a rectangle, that count turns out to be easy to get without laying out a single square by hand.

A rectangle, dragged by two opposite corners. A live reading counts how many unit squares it holds.

The count across and the count down multiply to the total every time you drag, on any rectangle, not just this one. Try it on a shape you haven’t dragged yet, before checking the number. That count is called the area of the shape — how many unit squares fit inside it. For a rectangle, the two side lengths that meet at a corner are its base (the same word an isosceles triangle’s third side used) and its height — the next lesson reuses both names unchanged for a right triangle’s two short sides.

Counting squares in two directions at once

Lay unit squares along the base: there are exactly as many as the base is long. Stack that same row up the height: there are exactly as many rows as the height is tall. The whole rectangle is that many rows of that many squares each — base times height, counted rather than assumed:

This is not a theorem being proved from something more basic. It is what multiplication already means, applied twice over, once along each side. Every area fact this course builds from here starts by comparing something to a rectangle: a triangle turns out to be half of one, a parallelogram turns out to equal one, and Pythagoras turns out to be a statement about three of them. Base times height is not one area formula among several — it is the one every other one is measured against.

A rectangle 9 units across and 5 units down.

A rectangle 12 units across and 3 units down.

Notice the second rectangle is longer and thinner than the first, yet holds fewer squares — area does not track how long a shape looks, only how many squares actually fit.

Do the exercise

This lesson counts as done once you get its exercise right. Nothing to tick off by hand.