Area
Area is the measure of the amount of space inside a two-dimensional shape, expressed in square units.
Formula
\text{Varies by shape; measured in square units}
Definition
Area tells you how much flat space is inside a shape, measured in square units like square centimeters ($\text{cm}^2$) or square feet ($\text{ft}^2$). Every shape has its own formula, derived from how it can be decomposed into unit squares or simpler shapes, ranging from a triangle's $(1/2)(\text{base})(\text{height})$ to a circle's $\pi r^2$, with composite shapes broken into pieces and their areas added. Formally, area is a measure on the sigma-algebra of Lebesgue-measurable subsets of $\mathbb{R}^2$: for regions bounded by smooth curves it equals the double integral of $1$ over the region, or by Green's theorem, $(1/2)\left|\oint (x \, dy - y \, dx)\right|$ over the boundary.
Example
If a classroom floor is $10$ meters long and $8$ meters wide, its area is $10 \times 8 = 80$ square meters, enough room to fit $80$ one-meter squares. A triangle with base $10$ cm and height $6$ cm has area $(1/2)(10)(6) = 30$ cm², and a circle with radius $5$ m has area $\pi \times 5^2 = 78.54$ m². The area enclosed by the ellipse $x^2/a^2 + y^2/b^2 = 1$ is $\pi a b$, generalizing the circle formula (where $a = b = r$); Green's theorem converts this area integral into a line integral, illustrating the link between area and boundary.
Key Insight
Area answers the question "how much surface?" while perimeter answers "how far around?": tiling a floor uses area, while putting baseboard around a room uses perimeter. All area formulas trace back to the rectangle (length $\times$ width): triangles are half a rectangle, parallelograms are rearranged rectangles, and circles are the limit of many thin triangles. The Banach-Tarski paradox shows that without measurability constraints, a sphere can be decomposed and reassembled into two spheres of the same volume, underscoring why measure theory needs careful axioms for area and volume.