Perimeter , the length of the boundary

The perimeter of a closed shape is the total length of its boundary. If you walked from a starting point all the way around the edge of the shape and returned to where you started, the distance you walked is its perimeter.

Concept

Perimeter is a length. Therefore it is measured in length units: centimetres (cm), metres (m), kilometres (km), inches, feet, and so on. Never in square units.

For shapes made of straight edges (polygons), the perimeter is just the sum of all the side lengths.

• A triangle with sides $5$ cm, $7$ cm, $4$ cm has perimeter $5 + 7 + 4 = 16$ cm.
• A square with side $s$ has perimeter $4s$.
• A rectangle with length $l$ and breadth $b$ has perimeter $2l + 2b = 2(l + b)$.
• A regular polygon with $n$ sides each of length $s$ has perimeter $ns$.

For irregular shapes , like the outline of a state on a map , there is no neat formula. You estimate by measuring small pieces with a string or a ruler and adding them up.

A useful real-world image: perimeter is "how much fence". If a farmer needs fencing for his rectangular field, he calculates the perimeter. If a town wants to put up Diwali lights along the boundary of a park, they measure the perimeter to know how much wire to buy.

Two shapes can have the same area but very different perimeters, and vice versa. Long thin shapes have lots of perimeter for their area. Compact shapes (closer to circles or squares) have less. We will see examples soon.

A common mistake

Perimeter is not the same as the number of sides. A square has $4$ sides, but its perimeter could be any number depending on side length. Always pay attention to the units.

Worked examples

Example 1. Find the perimeter of a triangle with sides $8$ cm, $10$ cm, $6$ cm.

• $P = 8 + 10 + 6 = 24$ cm.

Example 2. Find the perimeter of a square of side $7$ cm.

• $P = 4 \times 7 = 28$ cm.

Example 3. A rectangle is $12$ m by $5$ m. What is its perimeter?

• $P = 2(12 + 5) = 2 \times 17 = 34$ m.

Example 4. A regular hexagon (six-sided shape) has each side $9$ cm. Find the perimeter.

• $P = 6 \times 9 = 54$ cm.

Example 5. A rectangular garden has perimeter $40$ m. If its length is $12$ m, what is its breadth?

• $40 = 2(12 + b) \Rightarrow 20 = 12 + b \Rightarrow b = 8$ m.

Example 6. A wire of $48$ cm is bent to form a square. What is the side length? What if it is bent to form an equilateral triangle?

• Square: $4s = 48 \Rightarrow s = 12$ cm.
• Equilateral triangle: $3s = 48 \Rightarrow s = 16$ cm.

Try it yourself

1. Find the perimeter of a square with side $14$ cm.
2. Find the perimeter of a rectangle $18$ cm by $9$ cm.
3. A triangle has sides $12$, $14$, $13$. What is its perimeter?
4. A regular pentagon has perimeter $35$ cm. Find the length of each side.
5. A wire of $60$ cm is bent into a rectangle whose breadth is $10$ cm. Find the length.
6. The perimeter of an equilateral triangle is $66$ m. Find each side.
7. A field is shaped like an L. It is made of a $20$ m by $15$ m rectangle and a $10$ m by $5$ m rectangle joined at one edge. Sketch and find its perimeter.
8. Investigate: if you double the side of a square, what happens to its perimeter?

Activity

String measure. Find any irregular leaf, hand-print, or coin. Lay a piece of string around its edge, mark where it meets the start, and then measure the length of the string with a ruler. That length is the perimeter. Try at least three different shapes and compare.