Perimeter – The Distance Around a Shape
The perimeter of a shape is the total distance around its boundary. Think of it as the length of a fence needed to enclose a garden, or the amount of ribbon needed to go around a gift box.
What Is Perimeter?
Perimeter is measured in linear units (cm, m, km, etc.). To find it, add up the lengths of all sides. For curved shapes like circles, perimeter is called the circumference.
Perimeter Formulas
| Shape | Formula |
|---|---|
| Any polygon | Sum of all sides |
| Square (side s) | P = 4s |
| Rectangle (length l, width w) | P = 2(l + w) |
| Triangle (sides a, b, c) | P = a + b + c |
| Regular polygon (n sides, each s) | P = n × s |
| Circle (radius r) | C = 2πr |
| Semicircle (radius r) | P = πr + 2r |
Worked Examples
P = 2(8 + 5) = 2 × 13 = 26 cm.
P = 6 × 7 = 42 cm.
r = 5 cm. P = π(5) + 2(5) = 5π + 10 ≈ 15.71 + 10 = 25.71 cm.
2(11 + w) = 36. 11 + w = 18. w = 7 cm.
Compound Shapes
For shapes made by combining simpler shapes, trace the outer boundary carefully. Only include the sides that form the outer edge — internal divisions do not count.
Trace the outer boundary: 10 + 8 + 6 + 3 + 4 + 5 = 36 cm. (The two new inner edges add 3 cm + 4 cm but two original edges are shortened.)
Key Takeaways
- Perimeter = sum of all outer boundary lengths.
- Rectangle: P = 2(l + w). Square: P = 4s. Circle: C = 2πr.
- For compound shapes, only add the sides on the outer boundary.
- Always include units in your answer.
Practice Questions
- A square has perimeter 52 cm. Find its side length.
- Find the perimeter of a triangle with sides 7 cm, 9 cm, and 12 cm.
- A running track is a rectangle 120 m long and 60 m wide, with semicircles at each short end. Find the total distance around the track.
- A regular octagon has perimeter 72 cm. Find the length of one side.
- A rectangle has length (2x + 3) cm and width (x + 1) cm. Its perimeter is 38 cm. Find x and the dimensions.