Perimeter
Perimeter is the total distance around the outside of a two-dimensional shape, found by adding the lengths of all its sides.
Formula
P = \text{sum of all side lengths}
Definition
Perimeter is the distance all the way around the outside of a flat shape: you add up the lengths of every side. For a polygon this is simply the sum of all its side lengths, so a rectangle has $P = 2(l + w)$ and a regular n-gon with side length $s$ has $P = ns$. The idea extends to curved boundaries too: for a smooth curve $r(t) = (x(t), y(t))$ on $[a, b]$, the arc length (the perimeter analog) is $$L = \int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2} \, dt,$$ which reduces to the sum of Euclidean distances between consecutive vertices for polygons.
Example
A rectangle that is $5$ cm long and $3$ cm wide has a perimeter of $5 + 3 + 5 + 3 = 16$ cm, like the length of fence needed to go around a yard. An equilateral triangle with side length $7$ in has $P = 3 \times 7 = 21$ in, and a regular hexagon with side $4$ m has $P = 6 \times 4 = 24$ m; irregular polygons must have each side measured separately. The isoperimetric inequality shows why circles are special: among all closed curves of fixed perimeter $L$, the circle encloses the maximum area $A$, with $L^2 \ge 4\pi A$, equality holding only for circles.
Key Insight
The word "perimeter" comes from Greek: "peri" means around and "metron" means measure, so perimeter always measures length, never area, giving an answer in units like cm or feet rather than square units. Perimeter and area are independent of one another: two rectangles can share the same perimeter but hold very different areas, for example a $1 \times 9$ rectangle and a $3 \times 7$ rectangle both have perimeter $20$ but areas of $9$ and $21$. The isoperimetric problem, solved rigorously by Weierstrass, shows that nature favors circular shapes when minimizing boundary for a given enclosed area, which is why soap bubbles are spherical.