🟩 The counting game
The Square Counter
How big is a shape that refuses to be a rectangle? Everyone reaches for the same trick. Almost nobody knows the shortcut.
How big is a shape?
Nearly everyone starts by counting squares. One. By. One.
It works. It’s also slow. And there’s a way out.
The counting game
Count the squares that count
The classic move for a shape that won’t sit still: draw it on squared paper and count. A full square counts one. A square at least half covered counts one. Less than half? Nothing. The just-overs and the just-unders roughly cancel out.
Your turn. Tap every square that counts. The blob doesn’t care about your grid. That’s exactly what makes it interesting.
Counting every square is a tax.
Rectangles are where you stop paying it.
The shortcut
Nobody counts 108 squares
Counting works, and counting is slow. Drag the corner handle (or use the buttons) and stretch this one as big as it goes: the grid counts every square for you, live.
6 squares
3 across · 2 up
Now the suspicious part. A rectangle is rows: 2 rows with 3 squares in each. So 3 × 2 should give you the count with no counting at all. Decide first (same number, or not?), then check.
A rectangle is rows, stacked.
So a triangle should be hopeless… shouldn’t it?
The triangle trick
A right-angled triangle is half a rectangle
Triangles look like they should be miserable to measure: all those sliced squares along the slope. But every right-angled triangle sits inside a rectangle, and it is exactly half of it. Drag the corner to reshape the triangle, then complete the rectangle and watch the two numbers move together.
Straight edges give up easily.
Curves never do.
The awkward one
The circle that won’t count clean
Rectangles were easy. Triangles took one trick. Now try the circle. Same half-rule as before, but you choose how fine the grid is. Finer squares should mean a better answer. See how close it gets.
Closer, closer… never exact. However fine you cut, you cannot build a circle out of whole squares: the curved edge always cheats. The exact answer needs a famous number that hides inside every circle. It goes by a Greek letter, it never ends, and it is waiting for you in secondary school. When it turns up, act surprised.
Now the part people get wrong.
The fence is one measurement. The field is another.
Same length of fence. Different room inside.
The fence
Same fence, different carpet
Here is where most people go wrong: they mash area and perimeter into one idea. The two measure different things. Perimeter is the walk around the edge: a length, measured in cm. Area is the carpet inside: squares, measured in cm². This pen uses exactly 12 cm of fence, and the dot is walking it. Reshape the pen and watch which number moves.
Squares to count. Rows to multiply.
A fence to walk, a carpet to cover.
Two different measurements, and now you can tell them apart.
One more thing
This pair hides everywhere, and mixing them up costs real money. Skirting board is a walk: you buy it in metres. Carpet is a carpet: you buy it in square metres. A picture frame is a walk; the poster inside it is an area. A tin of paint doesn’t care how long your wall is, only how many square metres it has to cover. Once you see walk-numbers and carpet-numbers as two different measurements, you will see it everywhere.
And pass it on: next time someone finds the area of a rectangle by adding up the sides, you know what’s happened: they measured the fence and called it the field. Set them straight.