Measuring Space: Perimeter and Area · Lesson 2 of 10
Perimeter of a Circle and the Number π
“Measure a circle’s boundary and π appears, as usual, without ever finishing.”
• Understand the circumference-to-diameter ratio. • Define π and derive circumference formulas. • Appreciate major historical approximations of π. • Understand approximation versus equality. • Explain why π is irrational.
Circles can be very different in size. A small coin, a bicycle wheel, and a large circular track all have different circumferences and different diameters. As a circle becomes larger, both of these measurements increase, but something remarkable stays the same: the ratio of the circumference to the diameter.
If we divide the circumference of any circle by its diameter, we always get approximately the same value. This constant number is called pi, written as π, and its value begins 3.14159…. Because this ratio is the same for every circle, π allows us to connect a circle’s diameter or radius with its circumference and is one of the most important constants in mathematics.
π is the constant ratio of the circumference of any circle to its diameter.
Estimating π by Measurement
One practical method is to measure the diameter of a circular object, wrap a thin thread around it several times, divide the total thread length by the number of turns to estimate one circumference, and then calculate C/D. Repeating the measurement reduces the effect of small errors.
The Long Historical Search for π
Many civilisations developed increasingly accurate approximations. Mesopotamian mathematicians used about 3.125. Archimedes bounded π between fractions using inscribed and circumscribed polygons. Later, Ptolemy, Liu Hui, Zu Chongzhi, Āryabhaṭa and others improved the approximation.
| Source | Approximation / idea |
|---|---|
| Mesopotamian tradition | 3.125 |
| Archimedes | 3 10/71 < π < 3 1/7 |
| Ptolemy | 377/120 ≈ 3.14167 |
| Zu Chongzhi | 355/113 ≈ 3.1415929 |
| Āryabhaṭa | 3.1416, explicitly approximate |
| Mādhava | Infinite series approaching π exactly |
The important conceptual shift in Mādhava's work is that π is approached through an infinite process rather than represented by a single terminating computation or fraction.
Approximation Is Not Equality
The familiar 22/7 is useful, but it is not equal to π. The correct mathematical language distinguishes an approximation from an exact value.
π Is Irrational
A rational number can be written as a/b for integers a and b with b≠0. Its decimal expansion terminates or repeats. The digits of π continue forever without a repeating block, and π cannot be represented as a ratio of two integers. Therefore π is irrational.
A real number that cannot be written as a ratio a/b of integers with b≠0.
Practice Problems
- A circle has diameter 14 cm. Find its circumference using 22/7 as an approximation for π.
- A circle has radius 9 cm. Write its circumference exactly in terms of π.
- Explain the difference between π=22/7 and π≈22/7.
- Why does a repeating decimal correspond to a rational number, while π does not?
- Compare 22/7 and 355/113 as approximations of π using a calculator.
Key Takeaways
• C/D is the same for every circle and equals π. • Circumference is πd or 2πr. • π has a long history of increasingly accurate approximations. • An approximation is not an exact equality. • π is irrational and cannot be written exactly as a fraction of integers.
Next, we use circumference to measure arcs and explain the stagger on a 400 m track.