Perimeter
What is Perimeter?
Definition Perimeter
The perimeter of a shape is the total distance all the way around its outside edge.
Method Finding the perimeter
To find the perimeter of any shape, add the lengths of all its sides together.
Example
Find the perimeter of the red shape. Each square on the grid is 1 unit long.
- Step 1: Find the length of each side by counting the units on the grid.
- Step 2: Add the lengths of all the sides.$$\begin{aligned} \text{Perimeter} &= 2 + 2 + 2 + 2\\ &= 8\end{aligned} $$The perimeter of the shape is 8 units.
Perimeter of Common Shapes
Method Finding a Polygon’s Perimeter
To find the perimeter of any polygon (a closed shape with straight sides), add up the lengths of all its sides.
Proposition Perimeter Formulas
| Shape | Diagram | Perimeter Formula |
| Triangle |  | \(P = a + b + c\) |
| Rectangle |  | \(\begin{aligned}P &= l + w + l + w\\ &= 2l + 2w \\ &= 2(l + w)\end{aligned}\) |
| Square |  | \(P = s + s + s + s = 4s\) |
| Circle |  | \(P = 2\pi r\) |
Example
Find the perimeter of the rectangle:
The rectangle has a length \(l = 14\) m and a width \(w = 7\) m. We can use either perimeter formula.
- Method 1 (add both pairs of equal sides):$$\begin{aligned}[t]P &= 2 \times l + 2 \times w \\ &= 2 \times 14 + 2 \times 7 \\ &= 28 + 14 \\ &= 42 \,\mathrm{m}\end{aligned}$$
- Method 2 (factorise):$$\begin{aligned}[t]P &= 2 \times (l + w) \\ &= 2 \times (14 + 7) \\ &= 2 \times 21 \\ &= 42 \,\mathrm{m}\end{aligned}$$
Perimeter of Composite Figures
Definition Composite Figure
A composite figure is a shape made by joining two or more simple shapes (like rectangles, squares, and triangles).
Method Finding the Perimeter of a Composite Figure
To find the perimeter of a composite figure:
- Identify all the outer sides of the figure. Be careful not to include any lines inside the shape.
- Find the lengths of any unknown sides. You may need to use information from the other sides to figure these out.
- Add the lengths of all the outer sides together.
Example
Find the perimeter of this composite figure, which is made of a square and a triangle.
\(\quad\)
\(P=\)
+
\(\begin{aligned}[t]P &= \textcolor{colordef}{3\times 4 \mathrm{~cm}} + \textcolor{colorprop}{2\times 5 \mathrm{~cm}}\\P &= 22 \mathrm{~cm}\\\end{aligned}\)
\(P=\)
+\(\begin{aligned}[t]P &= \textcolor{colordef}{3\times 4 \mathrm{~cm}} + \textcolor{colorprop}{2\times 5 \mathrm{~cm}}\\P &= 22 \mathrm{~cm}\\\end{aligned}\)