Square a number and you're really laying tiles into a square. A square root just reads the picture backwards: what side made this square? Let's build a few and find out.
Start hereTo square a number means to multiply it by itself. So 5² (we say "five squared") is 5 × 5 = 25. It's called squared because 25 is exactly how many tiles fit in a square that's 5 along each side — a real, fillable square.
And a square root runs the same film backwards. When you write √25, you're asking one question: what side length made this square? Since a 5-by-5 square holds 25 tiles, the answer is 5 — so √25 = 5. Squaring builds the square; the square root reads off its side. That single back-and-forth is the whole page. Everything below is just you, some tiles, and that idea.
Here's the demo that makes "squared" stop being a word and start being a picture. Drag the slider to set the side — how many tiles run along one edge. The grid grows into a square of that size, and the readout counts every tile inside it. That tile count is the square number. Watch it: when the side is small, you can literally count the tiles (they're numbered); when the side is big, you trust the multiplication. Either way, side × side = total tiles.
Notice how the answer leaps, not creeps. Go from a side of 4 to a side of 5 and you don't add one tile — you add a whole new row and a whole new column, jumping from 16 tiles to 25. That's because you're multiplying, not adding. And here's the reverse already hiding in plain sight: the side of any of these squares is its square root. The 16-tile square has side 4, so √16 = 4. Read the picture one way and it's squaring; read it the other way and it's rooting.
Most maths words feel random. This one isn't. Long before anyone wrote 5² with a tiny floating 2, people were measuring area — how much flat space something covers — by counting unit squares. To find the area of a square tile floor, you don't measure cleverly; you just count the tiles. And a square that's 5 tiles wide and 5 tiles deep holds 5 × 5 = 25 tiles. The number you get by multiplying a length by itself was the area of a square, so people called it "squaring."
That's why the little raised 2 lives where it does. 5² means "the area of a 5-by-5 square," and the answer, 25, is a square number — a number of things you can arrange into a perfect square with none left over and no gaps. Try it with a handful of coins: 4 coins make a neat 2-by-2 square; 9 coins make a 3-by-3; but 7 coins refuse to square up no matter how you push them. The square numbers are special precisely because they fit.
"Squared" isn't a code word. It's a measurement. 5² is the area of a square that's 5 on every side — and the answer is a shape you could lay out on the floor.
If you square the counting numbers one after another — 1, then 2, then 3, and on — you get the perfect squares: the tile counts of squares with sides 1, 2, 3, 4… Here are the first twelve. They're worth getting friendly with, because they show up constantly. Tap any one to see it become a real square of dots, and to watch the multiplication that built it.
Look down the values and a sneaky pattern peeks out: the gaps between square numbers are the odd numbers. 1 to 4 is a jump of 3; 4 to 9 is a jump of 5; 9 to 16 is a jump of 7 — 3, 5, 7, 9, 11… That's the "extra row plus extra column" you saw a moment ago, minus the one corner tile they'd share. Patterns like this are everywhere once squares are tiles instead of just numbers.
Worth memorising the first dozen, the way you know your times tables: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Notice 100 = 10² and 144 = 12² — yes, the "gross" of 144 and the 12-by-12 multiplication grid are the same square number. Knowing these on sight makes square roots almost instant, which is exactly where we're headed next.
Squaring takes a side and gives you tiles. A square root does the opposite: it hands you a pile of tiles and asks you to find the side. Here's a target number of tiles. Your job is to rebuild it into a perfect square by dragging the side until it fits exactly — no tiles missing, none spare. The side you land on is the square root. Pick a target, then build it.
When your square holds exactly the target number of tiles, you've found the root: the side that fits is √(target). Because we chose perfect squares, there's always a whole-number side that works — that's what "perfect square" means. Spot the symmetry: √ and ² are an undo-pair, like adding and subtracting. √(7²) = 7, and (√49)² = 49. Do one, then the other, and you're right back where you started.
Perfect squares are the lucky ones — they have a tidy whole-number side. Most numbers don't. Try to build a square out of 10 tiles and you'll find a 3-by-3 (that's 9) is one tile short, while a 4-by-4 (that's 16) is six tiles too many. So √10 can't be a whole number — it has to live between 3 and 4. Slide along the number line below and watch where √N lands.
The ticks sit at the whole-number roots; the faint numbers beneath them are the perfect squares (1, 4, 9, 16…). When N is one of those, √N lands smack on a tick — a clean whole number. When N is anything else, the marker floats between two ticks, and √N becomes a never-ending, never-repeating decimal. √2 ≈ 1.41, √10 ≈ 3.16, √99 ≈ 9.95. You don't need to find those exact decimals by hand — but you can always say which two whole numbers a root sits between, and that's often all you need.
This is your first proper meeting with a kind of number that simply cannot be written as a neat fraction or a finishing decimal. √2 was famously the number that rattled the ancient Greeks, because no fraction on Earth squares to exactly 2. We call numbers like that irrational — not because they're unreasonable, but because they can't be written as a ratio of two whole numbers. Don't worry about taming the decimals now. The takeaway is gentler: every number has a square root, but only the perfect squares have a root you can write down exactly.
This is the single most common slip with squares, and nearly everyone makes it once. The little floating 2 looks like a "times 2," so your brain wants to read 5² as 5 × 2 = 10. But the raised 2 isn't a number you multiply by — it's a counter telling you how many 5s to multiply together. For squaring, that count is always two: the base, written down twice, and multiplied. Look at the two readings side by side.
Reading the little 2 as "times 2." Tempting, tidy — and wrong.
The 2 says "use two 5s." A 5-by-5 square holds 25 tiles.
A quick way to never fall for it again: picture the square. "Five squared" should make you see a 5-by-5 grid, not a pair of 5s being added or a single doubling. The gap between the two readings only widens as numbers grow: 10² is 100, not 20; 12² is 144, not 24. If your answer for a square ever comes out smaller than it feels like it should, check whether you accidentally multiplied by 2 instead of squaring.
Squaring and rooting aren't classroom-only tricks. Any time a shape's two directions are the same length — or any time you need to get a side back from an area — squares and their roots are doing the work.
The most famous rule in all of geometry runs on squares and roots. For a right-angled triangle, the rule a² + b² = c² says: square the two short sides, add them, and the square root of that total is the longest side. Every time someone finds a diagonal distance — across a screen, a football pitch, or a map — they're squaring, adding, and rooting. You'll meet this properly soon; squares and roots are the tools it's built from.
A square room 6 metres on each side has an area of 6² = 36 square metres — which is exactly how many 1-metre tiles you'd need. And if a tin of paint covers 36 square metres, you can root it back: a square wall of that area is 6 metres on a side.
A grid of pixels 100 wide and 100 tall holds 100² = 10,000 of them. Doubling a square image's width to 200 doesn't double the pixels — it quadruples them to 40,000, because both directions grow at once.
Lengths are measured in metres; areas in square metres — metres squared. The little ² in "m²" is the same squaring you've been doing all along. To get a length back from an area, you take a square root.
Square numbers help you check work and crack puzzles. If a quantity can be laid out as a perfect square — like 64 squares on a chessboard (8²) — that's a clue about its structure. Recognising 49, 64, 81 on sight saves real time.
No pressure, no marks recorded anywhere. Just check whether the tile picture has clicked. Pick an answer and you'll see straight away whether it fits, and why.
n² means n × n — the tiles in a square that's n on every side. 5² = 25. Never "5 × 2."
√ asks "what side made this square?" √25 = 5 because 5 × 5 = 25. They're an undo-pair.
Only perfect squares (1, 4, 9, 16…) have whole roots. √10 sits between 3 and 4.