Do you really need a complicated formula to find the area of a lumpy, irregular shape? Actually, no. With nothing more than a "geoboard" — a board of pegs where you stretch rubber bands to make shapes — you can find the area just by counting squares one by one. What makes a geoboard so appealing is that even a kid who hasn't learned the triangle area formula (base × height ÷ 2) yet can still find the exact area, purely by counting.
The method is simple. Count any square fully inside the shape as 1, and count any triangle-shaped square cut in half by a diagonal as 0.5. Counting this way, you can find the exact area of even the most complicated-looking shape. This works because every polygon you can make on a square grid can ultimately be broken down into a combination of "whole squares" and "squares cut in half."
There's also a more elegant way to find the area of a shape drawn on a grid of points, called Pick's theorem: area = (number of points inside the shape) + (number of points on the shape's edges) ÷ 2 − 1. Remarkably, the value this formula gives matches exactly what you get by counting squares one by one. This theorem, proved by the Austrian mathematician Georg Pick in 1899, shows that even area problems that look complicated can be turned into the very simple task of counting points.
The geoboard was originally invented in the 1950s by the British math educator Caleb Gattegno, as a tool that let students build shapes with their own hands. Using a real pegboard and rubber bands makes building, tearing down, and rebuilding shapes wonderfully free, letting you try out far more shapes than you comfortably could on paper. Not just triangles and quadrilaterals, but concave shapes and star shapes are all easy to build — and along the way, the idea of "area" stops being about plugging numbers into a formula and starts to feel like what it really is: how much space something takes up.
This square-counting approach connects naturally to the area formulas you'll learn later for shapes on a coordinate plane. Learn the area formulas for triangles or quadrilaterals, then check them again with a geoboard, and you'll realize, "oh, that formula gives the exact same answer as counting squares." Having this hands-on experience before memorizing the formula helps the formula feel far more meaningful once you get there.
When doing this activity with kids, it helps to prepare an actual pegboard and rubber band (or dot paper and colored pencils), give them a target area, and have them build as many different shapes matching that area as possible. Discovering that the same area can take the form of a triangle, a quadrilateral, a pentagon — completely different shapes — leads to a much more flexible understanding of what area really means.
On our activity page, you can build a shape by clicking points in order, as a puzzle where you try to hit a given target area exactly. Switch between Easy (5×5) and Hard (7×7) difficulty to build up your square-counting instincts.