A rotation turns a shape around a fixed point, like a hand sweeping around a clock. Choose the centre, the angle and the direction — and watch it swing.
Start hereA rotation takes a whole shape and turns it around one fixed point — without bending it, stretching it, or flipping it over. Think of the hands of a clock: they swing round and round, but they never change length.
That fixed point is the centre of rotation — the pin the shape swings around. It doesn't move at all; everything else does. To describe a turn completely you only need three things: where the pin is, how far around you turn (the angle), and which way — clockwise or anticlockwise. Get those three right and the shape lands in exactly one place.
By the end of this page you'll pin a centre wherever you like, spin a shape on a coordinate grid, read off exactly where every corner ends up, and describe any turn like a pro. No matrices, no scary formulas — just you, a shape, and a point to spin it around.
Every rotation is described with the same little handful of words. Learn them once and you'll read any question like a map.
A rotation is a turn: it spins a shape around a fixed point so that the shape stays exactly the same size and shape — it just faces a new direction. The fixed point it turns around is the centre of rotation. Picture pushing a drawing pin through the paper and swinging the shape around it; the pin stays put while everything swings.
The direction is either clockwise — the way a clock's hands move, from 12 toward 3 toward 6 — or anticlockwise, the opposite way (Americans say "counterclockwise"; it's the same thing). And the new, turned shape is called the image — the "after" picture. The shape you started with is the object, and mathematicians often mark a corner of the image with a little dash, so corner P becomes P′ (read "P prime") once it's been spun.
The pink pin is the centre of rotation. The dashed triangle is the object; the solid one is its image after a quarter-turn clockwise.
Corner P swings along the pink arc to land at P′. Notice P and P′ are the same distance from the pin — that's the secret we'll come back to.
This is the heart of the page, so play with it properly. On the grid below sits a shape (the ghosted, dashed one is the original — it never moves). Pick where to put the centre of rotation, choose an angle and a direction, then hit Spin. The solid pink shape is the image — where the shape lands after the turn.
Try the same angle clockwise and anticlockwise and watch the image land on opposite sides. Move the pin to a corner of the shape and spin — that corner stays put while the rest swings around it. Prefer the keyboard? Click the grid once to focus it, then use the arrow keys to nudge the pink pin around, and the buttons below to set the angle and direction. The readouts under the grid are the real, accessible answer.
P is the marked corner of the shape. Its image P′ is computed live from the centre, the angle and the direction.
You don't need a protractor for the neat turns. The three most common rotations — a quarter-turn (90°), a half-turn (180°) and a three-quarter turn (270°) — each move a point in a tidy, memorable way. Let's follow one corner, the point P(1, 1), and turn it around the origin (the point (0, 0) where the axes cross).
A quarter-turn the clock's way. The rule about the origin: the point (x, y) lands at (y, −x).
It drops from the top-right into the bottom-right.
A half-turn is the same clockwise or anticlockwise. The rule: (x, y) lands at (−x, −y).
Straight through the pin to the far side.
Three-quarters of the way round is the same as a quarter-turn the other way: (x, y) → (−y, x).
It swings up into the top-left.
Here's the picture behind those rules. Every corner of the shape stays exactly as far from the centre as it began — it just rides along a circle around the pin. A 90° turn moves it a quarter of the way around that circle, 180° moves it halfway (straight across, through the pin), and 270° moves it three-quarters of the way. That's why a half-turn looks the same whether you go clockwise or anticlockwise: halfway around a circle is halfway around, whichever direction you set off.
Now let's shift the pin off the origin. Take a 180° turn about the point (2, 1), and follow the corner P(4, 3). A half-turn sends every point straight through the centre to the same distance on the far side. P(4, 3) is 2 to the right of the centre and 2 above it, so its image is 2 to the left and 2 below:
Same trick, moved pin: measure how far the point sits from the centre, then flip that offset to the opposite side. That single idea — offset from the centre, then turned — is all a rotation ever does.
Why does a rotation never squash or stretch a shape? Because the centre keeps every point at the same distance for the whole turn. Drag the angle slider and watch: each corner traces a perfect circle around the pin, like a planet on its orbit or a hand sweeping a clock face. Corners far from the pin sweep a big circle; corners close to it sweep a tiny one — but nobody ever leaves their ring, so the shape arrives unbent and the same size.
The dashed rings are fixed. No matter how far you sweep, every corner stays glued to its own ring — so the shape keeps its exact size.
In a test you'll often be shown an object and its image and asked to describe the rotation fully. "Fully" is the key word — it means you must give all three ingredients, because leaving one out lets the shape land somewhere else:
The fixed point it turns around, written as coordinates — for example, "about (0, 0)" or "about (2, −1)". Miss this and the shape could pivot anywhere.
How far around: 90°, 180°, 270°, or any angle in degrees. This sets how big the turn is.
Clockwise or anticlockwise. You can skip it only for 180°, because a half-turn looks the same both ways.
So a complete answer sounds like: "a rotation of 90° anticlockwise about the point (0, 0)." Centre, angle, direction — the three-part sentence that pins a turn down to exactly one landing spot. A quick way to find the centre if you're not told it: join a point to its image and find the middle of that line for a 180° turn, or fold the paper so the object lands on its image — the crease points at the pin. But most of the time, the coordinates are handed to you, and you just read them off.
Time to prove it sticks. Each question turns a single point around the origin — work out where it lands and tap your answer. Remember the three tidy rules: 90° clockwise sends (x, y) to (y, −x); 180° sends it to (−x, −y); 270° clockwise (the same as 90° anticlockwise) sends it to (−y, x). Get a streak going.
Stuck on one? Picture the point on the grid, imagine the clock hand swinging it a quarter or half turn around the origin, and check which quadrant it ought to land in before you read the numbers. Direction only flips the 90° and 270° answers — the 180° answer is the same either way.
Rotation isn't just a page in a textbook — it's how a huge amount of the world moves and how a lot of design gets made.
Every clock hand, bike wheel and gear is a rotation about a fixed centre. The hand of a clock sweeps 360° in an hour, so it turns 90° — a quarter-turn — every 15 minutes. That's the exact same maths you just used on the grid.
Snowflakes, flowers, wheel rims and kaleidoscopes are built by rotating one small piece around a centre again and again. Designers call it rotational symmetry, and it's how a single petal becomes a whole bloom.
It runs deeper too. When a video game spins a character to face a new way, or a phone flips a photo upright, or an animator turns a wheel frame by frame, a rotation about a centre point is doing the quiet work behind the scenes. Even a Ferris wheel is just a giant rotation — each carriage riding its own circle around the middle, exactly like the corners of your shape.
Every turn swings around one fixed point — the centre of rotation. It never moves.
90°, 180°, 270° or any amount, clockwise or anticlockwise. Same size, new direction.
Centre, angle and direction — say all three and the image lands in exactly one place.