Interactive Geometry Lab — Hands-On Math Toys

Hands-on math · free · no sign-up

The Interactive Geometry Lab

A wall of math toys you can touch. Drag them, unfold them, slide them — every toy below runs right here in your browser, and each one lets you play a math idea until it clicks. No worksheets, no grades: just your hands and your curiosity.

Nine toys on the wall — each one has its own page now. Pick a card to open a toy on its own, or simply scroll down: all nine are playable right here on this page too.

Here for games instead? Build a personalised Math Adventure for your child, or flip cards together in the Memory Match Game.

1 · The Circle Trick — where πr² comes from

Textbooks hand you the formula for the area of a circle and ask you to memorize it. Here you can see it instead: cut a circle into slices, unfold the slices into a strip, and watch the strip turn into a rectangle as the slices get thinner. The rectangle’s area is easy — and it is exactly the circle’s area.

12
0%
length ≈ half way around the circle = π × r ≈ 15.7 height = r

Try it: first cut the circle into just 4 pieces and unfold — the strip is all wobbly. Now try 32 pieces. The more pieces you cut, the straighter the strip gets, until it is almost a perfect rectangle.

2 · Mirror Lab — symmetry you can drag

Butterflies, snowflakes and folded-paper cutouts all follow one rule: whatever sits on one side of the mirror line appears flipped on the other side, exactly as far away. Drag the orange dots and keep the blue twin honest — or switch to Free draw and invent a shape of your own.

Drag the orange dots — the blue shape is its mirror twin. Every dot stays exactly as far from the red line as its twin, on the other side.

In Free draw, click anywhere on the board to add a dot, drag dots to move them, and watch the mirror keep up. This is how butterfly wings, snowflakes and folded-paper cutouts work.

3 · Digital Huarong Dao — the sliding number puzzle

A beloved brain-teaser from China, named after the famous Three Kingdoms escape. One empty square, a board full of shuffled numbers, and one job: slide the tiles until the numbers run in order. Easy to learn, surprisingly hard to put down — and a real workout for planning-ahead muscles.

Moves: 0 Time: 0:00 Best (this board): —

The classic sliding-numbers game — in China it is called Digital Huarong Dao, after the famous Three Kingdoms escape puzzle. Slide the tiles into the empty space until the numbers run 1, 2, 3 … in order, with the empty space last.

Played enough? Put it into practice: head to our free math worksheets for printable and on-screen practice, or try a story at reading comprehension practice. Everything on this site is free, with no sign-up and no ads in your way.

4 · The Fraction Wall — fractions you can see

Ten bars, each one a whole cut into a different number of equal pieces. Paint pieces with a click and watch the total grow — then line two rows up and you can see that 1/2, 2/4 and 3/6 are the same amount wearing different clothes.

Every row below is one whole bar, cut into a different number of equal pieces. Click pieces to paint them.
Painted so far: 0 of one whole bar.
Try this: paint the 1/2 piece. Then, in the 1/4 row, paint pieces until you reach exactly the same spot. How many quarters did it take?

The magic of the fraction wall: fractions that end at exactly the same place are worth the same amount — 1/2 = 2/4 = 3/6 = 5/10. They are called equivalent fractions, and this wall lets you spot them with your eyes instead of memorizing rules.

5 · Make 24 — the card game

Four cards land on the table. Your job: use each card exactly once, with + − × ÷ and brackets, and land on exactly 24. It is a favourite brain-game in Chinese classrooms — and every hand we deal is checked solvable before it reaches you.

Rounds won: 0
Use each card exactly once, with + − × ÷ and brackets, and make exactly 24. Every hand we deal can be solved — we check before dealing.

This game (called "24 Points") is played in classrooms all over China. Fractions are allowed on the way — for example 8 ÷ (3 − 8 ÷ 3) = 24 — so do not give up if whole numbers alone do not work.

6 · Why Are Wheels Round?

Square wheels sound like a joke — until you watch one’s axle bounce. Roll both wheels and compare the trails their axles draw: one is a ruler-straight line, the other is a row of bumps. That trail is the whole secret of the round wheel.

Same size, same little platform on the axle. Very different ride.
Round wheel Square wheel

Press Play and watch the two trails left by the axles: blue for the round wheel, orange for the square one. Which platform would you rather ride on?

Nothing here is faked: the square wheel really tips from corner to corner, and its axle height is computed from the true geometry of the tipping square on every single frame.

7 · The Geoboard — shapes on a nail board

A board, some nails and one rubber band. Stretch the band into any polygon you like, and this board does something remarkable: it finds the area twice, by two completely different methods — measuring with coordinates, and just counting pegs (Pick’s theorem) — and the answers always match.

Click a peg to start your rubber band, then click more pegs to stretch it around. Click the first peg again (or press Close shape) when your polygon is finished.
No shape yet — your area, perimeter and peg counts will appear here once you close a shape.

A real geoboard is a board of nails and one rubber band — a classic primary-school tool in China for exploring shapes. The magic on this page is Pick's theorem: for any polygon whose corners sit on pegs, the area is hiding in two counts — the pegs inside the shape (I) and the pegs on its border (B). Area = I + B/2 − 1. We also measure the area a completely different way (the shoelace formula, which adds up the corners' coordinates) — and the two answers always agree. Try a slanted triangle: counting squares inside it is hard, but counting pegs is easy.

8 · The Dog and the Rope — how much grass can it reach?

A dog is tied to the corner of a square house. How much grass can it reach? It depends on the rope — and the moment the rope grows longer than one wall, the shape of the answer changes: the rope bends around the corners and two extra fans appear. This is a classic from the Chinese Grade 6 maths textbook.

5 m

This is one of the most famous problems in Chinese primary-school maths (it appears in the Grade 6 textbook, in the practice after the lesson on the area of a circle). The dog is tied to the corner of a square house. The house itself blocks one quarter of the circle, so with a short rope the dog sweeps a three-quarter fan. Drag the rope longer than one wall (L > 4 m) and watch what happens at the two corners: the rope bends around each corner and the leftover length sweeps out two extra quarter-circle fans on the far sides of the house. Past L = 8 m the fans would start to overlap each other, so the slider stops there.

9 · Cutting the Circle — Liu Hui’s road to π

How do you measure a circle with straight lines? You cheat: put a polygon inside it and keep doubling the sides. Around 263 CE the Chinese mathematician Liu Hui did this all the way to a 3,072-sided polygon and found π ≈ 3.1416. Here you can run his first five doublings yourself.

6 sides
SidesOne sidePerimeterπ ≈Misses π by

The circle's radius is 1, so its true circumference is exactly 2π ≈ 6.2832. Half the polygon's perimeter is therefore an estimate of π — computed live from the geometry (side = 2·sin(180°/n)), never typed in.

Nearly 1,800 years ago the Chinese mathematician Liu Hui (around 263 CE) computed π exactly this way — "cutting the circle" (割圆术). He started from a hexagon and doubled the sides again and again, all the way to a 3,072-sided polygon, reaching π ≈ 3.1416 with nothing but a ruler, counting rods and patience. His rule: "The finer you cut, the less is lost; cut and cut again until it cannot be cut — then it coincides with the circle and nothing is lost." Every number in the table above is his method, recomputed in your browser.

✓ Last checked November 2, 2026 — we re-test this tool every week to make sure it still works.