Area

Area is the measure of a two-dimensional region. It is always expressed in squared units (cm², m², ft²) because area is computed by multiplying two length dimensions.

Common formulas:

• Rectangle: $A = l \times w$ (length × width)
• Triangle: $A = \frac{1}{2} b h$ (half base × height) — Heron's formula handles three sides directly
• Circle: $A = \pi r^2$
• Trapezoid: $A = \frac{1}{2}(b_1 + b_2)h$
• Parallelogram: $A = b h$
• Regular polygon: $A = \frac{1}{2} P a$ (half perimeter × apothem)

Calculus generalises area to integration: $\int_a^b f(x)\,dx$ is the signed area between $y = f(x)$ and the x-axis on $[a, b]$. This is how we compute the area of any region bounded by curves, not just classic shapes.