Measuring Space: Perimeter and Area · Lesson 1 of 10
Perimeter of a Shape
“Perimeter is what happens when a shape asks you to walk all the way around it.”
• Understand perimeter as total boundary length. • Recall perimeter formulas for common polygons. • Understand why circular perimeter is called circumference. • Recognise proportional relationships between size and perimeter. • Connect perimeter with real situations such as running tracks.
Perimeter
Perimeter tells us the total distance around the outside boundary of a shape. To find it, we imagine starting at one point on the boundary and moving all the way around the shape until we return to the same point. Every side or curved part of the boundary contributes to this total distance.
A simple way to picture this is to imagine a tiny insect walking along the edge of the shape without cutting across the inside or turning back. The complete distance covered in one full trip around the boundary is the perimeter. This is why perimeter is measured in units of length, such as centimetres, metres, or kilometres.
The total length around the boundary of a two-dimensional shape.
| Shape | Side information | Perimeter |
|---|---|---|
| Square | side a | 4a |
| Equilateral triangle | side a | 3a |
| Rectangle | length a, width b | 2(a+b) |
The square formula is actually a special case of the rectangle formula. If a rectangle has equal sides, b=a, then 2(a+b)=2(a+a)=4a.
Why Athletics Tracks Need a Stagger
On a running track, athletes in outer lanes travel along curves with larger radii. The straight portions may be equal, but the curved parts are longer in the outer lanes. To make every runner cover the same total distance, the outer lanes must begin slightly farther ahead. This starting offset is called the stagger.
Proportional Change in Perimeter
If the side of a square doubles, its perimeter doubles. If the side triples, its perimeter triples. This happens because the perimeter is directly proportional to the side length.
For an equilateral triangle the same idea gives P/a=3. The ratio remains fixed even when the shape is enlarged or reduced.
What About a Circle?
A circle has no straight sides, so we cannot add side lengths. Its perimeter has a special name: circumference. The important question is whether circumference also has a fixed ratio to some basic measurement of the circle.
The perimeter, or total boundary length, of a circle.
Practice Problems
- Find the perimeter of a rectangle of length 18 cm and width 7 cm.
- A square has perimeter 52 cm. Find its side length.
- An equilateral triangle has perimeter 45 cm. Find its side.
- Explain why a runner in an outer lane covers a longer curved distance than a runner in an inner lane.
- If every length of a square is multiplied by 3, what happens to its perimeter?
Key Takeaways
• Perimeter is the total distance around a boundary. • Circumference is the perimeter of a circle. • Perimeter often changes directly with a shape's linear dimensions. • Running-track staggers compensate for different curved-path lengths. • The next step is to study the constant circumference-to-diameter ratio.
Next, we explore the C/D ratio, the history of π, and why π is irrational.
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Perimeter of a Circle and the Number π