Area

Counting squares

Area is how many unit squares fit inside a shape — squares one unit wide and one unit tall.

$$A_{\text{rectangle}} = l \times w \qquad A_{\text{triangle}} = \tfrac{1}{2}bh \qquad A_{\text{circle}} = \pi r^2$$

Every one of those multiplies two lengths, which is the common thread.

Why the units are squared

The exponent counts the dimensions.

Multiplied	Quantity	Units
one length	perimeter	cm
two lengths	area	cm²
three lengths	volume	cm³

This is the fastest check on any measurement answer. A number reached by adding lengths cannot carry square units, so “perimeter = 24 cm²” is impossible at a glance.

Area and perimeter are independent

They vary almost freely against each other.

Rectangle	Perimeter	Area
$1 \times 11$	$24$	$11$
$6 \times 6$	$24$	$36$

Same fence, triple the field. Perimeter is the fence; area is the field.

Area adds

An awkward shape can be cut into pieces and its area is the sum of them, provided the pieces do not overlap. That is why an L-shaped room needs no formula of its own — see area of a rectangle.

Surface area

The area of the outside of a solid — how much wrapping paper covers it. Still in square units, because it is still area. Not to be confused with volume, which counts what fits inside and uses cubic units.