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.
Legend tells of Queen Dido of Carthage, who was granted as much land as she could enclose with a single ox-hide; she is said to have cut the hide into thin strips, tied them into one long cord, and stretched it into a huge semicircle against the coastline — instinctively solving what mathematicians now call the isoperimetric problem: for a fixed perimeter, a circle (or a semicircle against a straight edge) always encloses the greatest possible area. That same relationship between boundary length and enclosed space still matters today: an Olympic running track has a standard inner-lane perimeter of exactly 400 metres, carefully designed with straight sections and curves so every runner in every lane covers a fair, precisely measured distance.
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.