Online Geoboard

Give Maya four pegs at the corners of a 1-by-1 square. Her rubber band travels 4 board units around the outside and covers 1 square unit inside. Move one corner two pegs away and both numbers change in front of her. That immediate link between a movement, a shape, and two measurements is the mathematical payoff of a geoboard.

Perimeter is the easier number to see. The tool measures the straight distance from each corner to the next and includes the closing edge back to the first peg. On a square grid, horizontal and vertical neighbours are 1 unit apart. A diagonal across a unit square is about 1.41 units because the shortest path is a straight line, not a walk around two sides.

Area needs a more powerful rule because counting whole grid squares stops working when edges are diagonal or the shape turns inward. The tool uses the shoelace formula: write the corner coordinates in order, multiply across each neighbouring pair, subtract the reverse products, and divide the absolute result by 2. A triangle at (0,0), (3,0), and (0,4) gives area 6 square units. A concave L-shape also works without cutting it into rectangles first.

The formula depends on one clean boundary, so the tool prevents a band from crossing itself. Separate shapes may overlap freely. Their translucent fills show the shared region while each band keeps its own area and perimeter, which makes comparison work possible without erasing the first example.

The board layout changes the questions students notice. Pick the layout for the idea you want to teach, not simply for the age of the class.

• 5x5 square: the spacing is generous on a tablet and every horizontal or vertical gap is one unit. It is the clearest choice for first work with rectangles, right triangles, fractions of a square, and Pick's theorem investigations.
• 10x10 square: the unit rule stays the same, but 100 pegs allow compound shapes, scaled copies, coordinate challenges, and several overlapping solutions. Use it when the 5x5 board forces every answer to look alike.
• Isometric: alternate rows are offset by half a unit and the vertical spacing is √3/2, making every nearest-neighbour gap exactly one unit. Equilateral triangles, rhombi, hexagons, and 60-degree turns appear naturally. Areas often include decimals because an equilateral unit triangle has area about 0.43 square units.
• Circular: eight pegs sit on a ring of radius 2, sixteen on a ring of radius 4, and one sits at the centre. It supports symmetry, chords, inscribed polygons, central angles, and questions about how equal-radius points can still create different side lengths.

Shapes remain saved separately on each layout while the page stays open. A teacher can prepare one square-board example and one circular-board example, switch between them during discussion, and return without rebuilding either board.