Area and Perimeter — Mathematics Knowledge System

Overview

Perimeter is the total distance around the boundary of a two-dimensional shape; area is the measure of the region enclosed by that boundary — how much surface it covers. Area can be thought of as the amount of paint needed to cover a shape with a single coat, while perimeter is the length of fencing needed to enclose it. Every standard polygon and the circle has a well-known formula for each, derived from decomposing the shape into simpler pieces (typically triangles and rectangles) or, for the circle, from a limiting process.

Formal Definition

Definition

Standard area and perimeter formulas for the three most common plane figures:

A_{\triangle} = \tfrac{1}{2}bh, \qquad P_{\triangle} = a+b+c

Triangle: area (base × height / 2) and perimeter (sum of sides)

A_{\text{rect}} = lw, \qquad P_{\text{rect}} = 2(l+w)

Rectangle: area (length × width) and perimeter

A_{\circ} = \pi r^2, \qquad C_{\circ} = 2\pi r

Circle: area and circumference (perimeter)

Worked Examples

Area is length times width.
$A = 8 \times 5 = 40$
Perimeter is twice the sum of length and width.
$P = 2(8+5) = 26$

Answer: A = 40, P = 26

Practice Problems

Difficulty 2/10

A circle has radius 7. Find its area and circumference (use π ≈ 3.14).

Summary

Perimeter measures the distance around a shape's boundary; area measures the region it encloses.
Triangle: A = ½bh, P = a+b+c. Rectangle: A = lw, P = 2(l+w). Circle: A = πr², C = 2πr.
Complex polygons are typically found by decomposing into triangles and rectangles and summing.
Units matter: perimeter/circumference are linear units (e.g. cm), area is always in squared units (e.g. cm²).

References

WebsiteWikipedia — Area