Circumference, area of a circle and the number π

What is it about?

For every circle the circumference $C$ divided by the diameter $d$ is the same number: $C : d = \pi \approx 3.14159\ldots$ , so $C = \pi \cdot d = 2\pi \cdot r$ and $A = \pi r^2$ . Rolling a wheel once covers a little more than three diameters. Cutting a circle into many sectors and rearranging them gives almost a rectangle with height $r$ and width $\pi r$ . Archimedes trapped π between polygons: $3\frac{10}{71} < \pi < 3\frac{1}{7}$ .

Try it

Press Roll and compare with the diameters below.
Rearrange 48 sectors. 48 sectors
Double the corners like Archimedes. Hexagon

Exercises

Exercise 1: Bicycle wheel

A wheel has a diameter of $70\,\text{cm}$ . How many turns are needed for $1\,\text{km}$ ?

Show solution

$C = \pi \cdot 70\,\text{cm} \approx 219.9\,\text{cm} \approx 2.2\,\text{m}$ , so $1000\,\text{m} : 2.199\,\text{m} \approx 454.7$ , about $455$ turns.

Exercise 2: One big or two small pizzas?

Compare one pizza with $d = 30\,\text{cm}$ and two with $d = 20\,\text{cm}$ .

Show solution

$707\,\text{cm}^2$ versus $628\,\text{cm}^2$ – the big one is larger.

Ideas for the classroom

Use: Grade 8 (circumference, area of a circle, approximations of π).