Perimeter
Geometry & MeasurementPerimeter 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. To find it, you add up the lengths of every side.
Example
A rectangle that is $5$ cm long and $3$ cm wide has a perimeter of $5 + 3 + 5 + 3 = 16$ cm. Think of it as the length of fence needed to go around a yard.
Key Insight
The word "perimeter" comes from Greek: "peri" means around and "metron" means measure. Perimeter always measures length, so the answer is always in units like cm or feet, never square units.
Definition
The perimeter of a polygon is the sum of the lengths of all its sides. For a rectangle, $P = 2(l + w)$. For a regular n-gon with side length $s$, $P = ns$.
Example
An equilateral triangle with side length $7$ in has $P = 3 \times 7 = 21$ in. A regular hexagon with side $4$ m has $P = 6 \times 4 = 24$ m. For irregular polygons, each side must be measured separately.
Key Insight
Perimeter and area are independent: two rectangles can have the same perimeter but very different areas. A $1 \times 9$ rectangle and a $3 \times 7$ rectangle both have perimeter $20$, but areas of $9$ and $21$.
Definition
For a smooth curve in the plane parameterized by $r(t) = (x(t), y(t))$ on $[a, b]$, the arc length (perimeter analog) is $$L = \int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2} \, dt.$$ For polygons, this reduces to the sum of Euclidean distances between consecutive vertices.
Example
The isoperimetric inequality states that among all closed curves of fixed perimeter $L$, the circle encloses the maximum area $A$, with the relation $L^2 \ge 4\pi A$. Equality holds only for circles.
Key Insight
The isoperimetric problem, solved rigorously by Weierstrass, shows that nature favors circular shapes when minimizing boundary for a given enclosed area, explaining why soap bubbles are spherical.