The Dog and the Rope — how much grass can it reach?
A dog tied to the corner of a house, a rope, and one of the most famous problems in Chinese primary-school maths.
Tie a dog to the corner of a square house and ask: how much grass can it reach? With a short rope the answer is a three-quarter circle, because the house blocks the rest. But let the rope grow longer than one wall and the problem transforms: the rope bends around the corners, and the leftover length sweeps out two extra fans on the far sides of the house. Generations of Chinese sixth-graders have met this problem in their textbook.
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.
How to play
Drag the rope-length slider from 1 to 8 metres and watch the reachable region redraw itself, with the area formula updating piece by piece. Notice the exact moment the rope passes 4 metres — one wall — and the two small fans appear.
What you learn
You learn to split a complicated region into simple fans, and to notice when a growing quantity changes the shape of a problem, not just its size. It is the area of a circle, wearing a story.
Want more? Browse the whole Interactive Geometry Lab for nine hands-on math toys, or build a personalised math adventure for your child with the free Game Builder. Everything is free, with no sign-up.
✓ Last checked November 2, 2026 — we re-test this tool every week to make sure it still works.