Geoboard: Free Online Geoboard With Area and Perimeter

What a geoboard teaches

A geoboard turns geometry into something you can stretch. A child puts a rubber band around four pegs and has a square. Moving one corner one peg makes a different shape, and the child sees exactly what changed: a side got longer, an angle opened up, the area grew by half a square. That quick feedback between action and result is why teachers have used geoboards for decades.

A geoboard lets learners compare polygons by moving corners between fixed pegs. Count unit squares for a first area model, then explain a formula using rectangles and triangles.

On a square board, one square between four neighboring pegs is one square unit. Everything else is measured from that: a 2 by 3 rectangle covers 6 squares, a triangle that is half of that rectangle covers 3, and a tilted square can cover 2 or 5.

How to use this online geoboard

Drag a band from the tray onto the pegs, then drag its corners. Each corner is a dashed ring that snaps to the nearest peg. To move a whole band, press Select band and drag it, or drag across the corners to select them.

The readout follows the band you touched last:

• Shape: the name, from "right isosceles triangle" to "parallelogram", "kite" or "concave hexagon".
• Area: exact, in square units. Areas on a geoboard are always whole numbers or halves.
• Perimeter: the sum of the sides, kept exact. A square tipped on its corner has perimeter 4√2, shown with its decimal, about 5.66.
• Border and inside pegs: the numbers B and I that Pick's theorem uses.

Turn on Grid squares to draw faint unit squares between the pegs, which helps young children count area.

Area and perimeter on a geoboard

Area and perimeter are easy to confuse, and the geoboard pulls them apart. Make a 1 by 4 rectangle and a 2 by 2 square. Both have area 4, but the rectangle's perimeter is 10 and the square's is 8. Then make a 2 by 3 rectangle: perimeter 10 again, but area 6. Children who build these see that two shapes can share one measurement and differ in the other.

The Common Core standards teach area by counting unit squares in 3rd grade (3.MD.C.5 to C.7) and perimeter in the same year (3.MD.D.8), including "rectangles with the same perimeter and different areas." In 6th grade, students find the area of triangles and other polygons by composing rectangles or splitting shapes into triangles (6.G.A.1). The geoboard supports both: count squares at first, then box a slanted shape in a rectangle and subtract the corner triangles.

Pick's theorem: area from counting pegs

In 1899 the mathematician Georg Pick proved a surprising shortcut. For any shape whose corners sit on pegs and whose sides don't cross:

Area = I + B ÷ 2 − 1

where I is the number of pegs inside the band and B is the number of pegs the band touches, corners included. A 2 by 2 square has 1 peg inside and 8 on the band, so its area is 1 + 4 − 1 = 4. The readout shows the theorem worked out for every band, so older students can predict an area, check it, and then wonder why it works. Task card 7 asks for a shape with 6 border pegs and 2 inside pegs; Pick's theorem says its area will be 4 before you build it.

Tilted squares and the Pythagorean theorem

Ask a child to make a square with area 2 and they will usually say it's impossible: a 1 by 1 square is too small and a 2 by 2 square too big. Then tip the square onto its corner, with each side going 1 across and 1 up. Its area is exactly 2, so each side is √2 long.

The same trick makes a square of area 5 (sides 2 across and 1 up) and area 10 (3 across and 1 up). The pattern is the Pythagorean theorem: a side that goes a across and b up has length √(a² + b²), and the tilted square on it has area a² + b². In the Common Core standards, 8th graders apply the theorem to find distances on a grid (8.G.B.8). The geoboard lets younger children discover the idea by building it.

The circle geoboard

Switch to the circle board for angles. It has 12 pegs around a circle, 30° apart, plus a center peg. Every angle a band makes there is a multiple of 15°, and the readout lists each one with their sum.

• Equilateral triangle: every fourth peg. All three angles are 60°.
• Square: every third peg. All angles are 90°.
• Regular hexagon: every other peg. All angles are 120°, and the area is 3√3/2 when the radius is 1.
• Right angle from a diameter: stretch a band across the middle, then add a third corner anywhere on the circle. The angle at the third corner is always 90°, a fact known as Thales' theorem.

Geoboards by grade

Activities to try

• All the triangles with area 1. How many differently shaped triangles with area exactly 1 can you make on the 5 × 5 board?
• Same area, different shapes. Make three shapes with area 6 that look nothing alike.
• Largest square. What is the biggest square that fits on the 5 × 5 board? Is a tilted one ever bigger? (On a 5 × 5 board, no: the straight 4 by 4 square, area 16, is the largest.)
• Shape riddles. "I have four sides, one pair of parallel sides and a right angle." Build it, then check the shape name in the readout.
• Fold a design. Make a shape on one half of the board and its mirror image on the other half, using a second band color.
• Predict, then build. Before moving a corner, predict how the area will change. Move it and compare.

Tips for teachers and parents

• Count first, then check. Hide the value, let children count the area, then show the value. Use the readout to check an explanation, not to replace the learner's reasoning.
• Use dot paper as the record. After building a shape, children draw it on dot paper and label its area and perimeter.
• Talk about exact lengths. When a side is √5, resist rounding it. The exact value is what makes the tilted squares work.
• Use colors on purpose. Put the original shape in red and a changed version in blue, then compare the two readouts.

What makes this geoboard different

Most free geoboards stop at drawing bands. This one measures them exactly: the area as a whole number or half, the perimeter as a sum of simplified square roots, the pegs on and inside the band with Pick's theorem worked out, and every angle on the circle board. It adds three boards, six band colors, side lengths drawn on the board, undo and redo, share links, a printable snapshot, keyboard controls and twelve self-checking task cards.

Want more geometry? Plot shapes with coordinates on the interactive coordinate plane, or build shapes from pattern blocks. Print graph paper for area problems on paper.

Choosing an activity

Grade bands describe possible curriculum connections, not an assessment of an individual learner. Start with an example the learner can explain, then change one feature at a time. The activity suggestions here are editorial teaching ideas, not results from a classroom trial of this software. Use the model alongside written work and physical materials when useful. The linked standards describe learning objectives; they do not endorse Edulist or require this particular tool.