Area and perimeter

Perimeter and area are the two basic measurements of any two-dimensional figure. The perimeter is the distance around the shape — what you would walk if you traced its boundary — found by adding the lengths of all its sides. The area is the amount of surface the shape encloses. Perimeter is measured in linear units like feet or centimeters; area is measured in square units like square feet or cm².

A handful of formulas cover most problems. A rectangle with length l and width w has perimeter P = 2l + 2w and area A = lw; a square with side s has P = 4s and A = s². A triangle's area is A = (1/2)bh, using the base and the height perpendicular to it. For circles the perimeter is called the circumference: C = 2πr, with area A = πr². For example, a 5 × 3 rectangle has perimeter 2(5) + 2(3) = 16 and area 5 × 3 = 15 square units.

The two measures are independent — shapes with the same perimeter can enclose very different areas. A 4 × 4 square and a 7 × 1 rectangle both have perimeter 16, but their areas are 16 and 7. Test writers exploit exactly this: composite figures, shaded regions found by subtracting one area from another, and problems that give area and ask for perimeter (or vice versa) all reward knowing which measure is which.

Area and perimeter questions appear throughout the geometry sections of the ACT, the AMC math competitions, and the CLT. Memorize the core formulas, keep your units straight (linear vs. square), and practice breaking irregular figures into rectangles, triangles, and circles you can handle piece by piece.