Perimeter

Perimeter is the total length around a two-dimensional figure.

Boundary, not interior

Imagine walking once around a garden’s edge. The distance travelled is its perimeter. Area measures the interior surface instead, so perimeter uses linear units such as metres, while area uses square metres.

The general method is dependable for every polygon: identify every outside edge and add its length exactly once.

Common formulas

For a rectangle of length $l$ and width $w$,

For a square with side $s$, $P=4s$. A regular $n$-gon with side length $s$ has $P=ns$. Formulas compress repeated addition; they do not replace understanding which edges form the boundary.

A rectangle example

The measurement answer is $29$ m; the purchasing decision requires an additional interpretation.

Missing side lengths

Diagrams often omit a side that can be inferred. Opposite sides of a rectangle are equal. In an L-shaped figure, aligned horizontal segments must total the same overall width, and aligned vertical segments must total the same overall height.

Label inferred lengths before adding. This separates geometric reasoning from arithmetic and makes accidental omissions easier to spot.

Composite and irregular shapes

Trace the outside boundary with a finger or pencil, starting at one vertex and returning to it. Create an ordered list of edge lengths. Interior divider lines are not part of the perimeter unless the context specifically includes them.

For curved pieces, include arc length rather than a straight shortcut. A semicircle with radius $r$ has curved length $\pi r$; if its diameter is part of the boundary, add $2r$ as well.

Circumference

The perimeter of a circle has the special name circumference:

Keep $\pi$ for an exact result or round only at the end. The diameter is twice the radius, so choose the formula matching the given measurement.

Scale and similarity

If every length is multiplied by scale factor $k$, perimeter is also multiplied by $k$. A model enlarged by factor $3$ has three times the perimeter—not nine times. The square factor $k^2$ belongs to area.

Units and precision

All lengths must use the same unit before addition. Convert $2.4$ m and $85$ cm to either metres or centimetres, not a mixture. Measurement precision also matters; a perimeter calculated from rounded dimensions is itself approximate.

Common mistakes

Using an area formula. Perimeter adds boundary lengths and stays in linear units.

Counting interior lines. Only the outside edge counts unless the situation says otherwise.

Missing an unlabelled side. Use shape properties and aligned totals to infer it.

Mixing units. Convert first, then add.

Quick self-check

• Have I traced one complete trip around the boundary?
• Is every outside segment counted once?
• Are all measurements in the same linear unit?
• Is the result reasonable compared with the figure’s dimensions?

Related topics

• Polygons
• Area
• Circumference
• Surface Area