Perimeter, Circumference, and Area

Perimeter is just a walk around the edge: add up the side lengths. For a circle, that walk is the circumference, found with C = 2 π r (or π d). Area measures what's inside, and it behaves completely differently. A rectangle's area is length times width, and a circle's is π r^2. You can see where π r² comes from by slicing a circle into thin wedges like a pizza and laying them alternately pointing up and down. They form a shape close to a parallelogram with height r and base equal to half the circumference, π r. Multiply those: π r times r gives π r^2.

This is why two rectangles can share a perimeter but hold very different areas. A 1 by 11 rectangle and a 6 by 6 square both have perimeter 24, but areas of 11 and 36. Perimeter only tracks the boundary; area tracks how the space is arranged inside it. That same distinction shows up when a shape is built from pieces, like an L-shaped floor made by cutting a smaller rectangle out of a larger one: you find each rectangle's area separately and subtract, since area adds and subtracts piece by piece even when perimeter doesn't work that simply.

Scaling ties it together. Stretch every side of a rectangle by a factor of 2, and the area doesn't double, it quadruples, because area depends on two dimensions multiplied together. This squared relationship between length scale and area scale is worth remembering anywhere shapes are resized, from floor plans to circles, where doubling the radius quadruples π r² too.