Area and Perimeter Builder
Draw shapes on a square grid and watch the area and perimeter count up. Challenges ask for shapes with the same area but different perimeters.
Area
Perimeter
How to use it
Click squares to fill them, or drag to fill several; start a drag on a filled square to rub out. The area counts the filled squares and the perimeter counts the unit edges around the outside, drawn in red. On a keyboard, click the grid or tab to it, move with the arrow keys and press Space. Rectangles mode draws a rectangle from a length and width and shows the formulas.
The three challenges are checked by the tool: make a shape with a given area and perimeter, make two shapes with the same area but different perimeters, or find the smallest perimeter for an area. Shapes must be one piece, joined edge to edge.
The math: area covers, perimeter goes around
Area is how many unit squares cover a shape, measured in square units. Perimeter is the distance around it, measured in units. For a rectangle, area = length × width and perimeter = 2 × (length + width). For any other shape made of squares, counting still works: every edge of a square that does not touch another filled square is part of the perimeter. If a shape has a hole, the edges around the hole count too, because they are also a boundary.
Shapes with the same area can have very different perimeters. Twelve squares in a 3 by 4 rectangle have a perimeter of 14, but in a long line of 12 the perimeter is 26. Squarer shapes have the smallest perimeter for their area, which is why animals curl up into a ball to keep warm, and stretched, frilly shapes have the largest, like a leaf or a lung that needs lots of edge.
- Can two shapes have the same perimeter but different areas? Find a pair.
- Add one square to a shape. Can the perimeter go up by 2, stay the same, or go down?
- Why is the perimeter of a shape made of squares always an even number?
Free to use, no account needed. It runs in your browser: nothing you type, and no sound from your microphone, is sent to Fieldbook.
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