A worked example: circle with radius 5
A circle with radius 5 has area π × 5² = π × 25 = 78.539816 square units. The radius is squared first, then multiplied by π, because area scales with the square of a linear dimension, not the dimension itself.
The formula: four shapes, four formulas
A rectangle 8 units long and 6 units wide has area 8 × 6 = 48 square units — length times width. A square with side 7 has area 7² = 49 square units, since a square is just a rectangle whose length and width are equal. A triangle with base 10 and height 6 has area 10 × 6 ÷ 2 = 30 square units — half the rectangle you'd get from the same base and height, because the triangle only covers half of that rectangle.
Another worked example: a bigger rectangle
A rectangle 12 units long and 6 units wide has area 12 × 6 = 72 square units. Doubling the length from 6 to 12 while keeping the width at 6 exactly doubled the area from 36 to 72, which is what "area scales linearly with each individual dimension" means in practice.
Why the units are always squared
Every one of these formulas multiplies two lengths together (or a length by itself, or a length by π), which is why area is always reported in square units — square meters, square feet, and so on — never in plain linear units. This is also why doubling a single dimension only doubles the area, but doubling every dimension of a shape quadruples it.