Area
Geometry & MeasurementArea 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. It is measured in square units, like square centimeters ($\text{cm}^2$) or square feet ($\text{ft}^2$).
Example
If a classroom floor is $10$ meters long and $8$ meters wide, its area is $10 \times 8 = 80$ square meters. You could fit $80$ one-meter squares on that floor.
Key Insight
Area answers the question "how much surface?" while perimeter answers "how far around?" Tiling a floor uses area; putting baseboard around a room uses perimeter.
Definition
Area is the measure of the two-dimensional region enclosed by a figure, expressed in square units. Each shape has its own formula derived from how it can be decomposed into unit squares or simpler shapes.
Example
A triangle with base $10$ cm and height $6$ cm has area $(1/2)(10)(6) = 30$ cm². A circle with radius $5$ m has area $\pi \times 5^2 = 78.54$ m². Composite shapes are broken into simpler pieces and their areas added.
Key Insight
All area formulas can be traced back to the rectangle (length $\times$ width). Triangles are half a rectangle; parallelograms are rearranged rectangles; circles are the limit of many thin triangles.
Definition
Area is a measure on the sigma-algebra of Lebesgue-measurable subsets of $\mathbb{R}^2$, assigning a non-negative real number to each region. For regions bounded by smooth curves, the area 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
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
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 requires careful axioms for area and volume.