Geoboard Area Explorer — stretch a band, find the area twice
Nails, a rubber band and a small miracle: the area of any polygon, found by counting pegs alone.
A geoboard is a board of nails in a grid, with one rubber band to stretch around them. Any polygon you can stretch, you can measure — and this explorer measures it twice. First it computes the area the school way, from coordinates, using the shoelace formula. Then it simply counts: pegs inside the shape, pegs on its border, and one line of arithmetic called Pick theorem. The two answers always agree.
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.
How to play
Click pegs one by one to stretch the band, then click the first peg again (or the Close shape button) to finish. The board reports the area, the perimeter, the border pegs and the inside pegs. Press the example button to see a worked triangle first.
What you learn
Pick theorem says area equals inside pegs plus half the border pegs, minus one — a result so surprising that checking it against the shoelace answer feels like a magic trick. Along the way you practise coordinates, perimeter, and the idea that area counts unit squares.
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.