The circumference, the area of a disc and the number π

The circumference is $\displaystyle \pi$ times the diameter, and the area of a disc is $\displaystyle \pi$ times the square of the radius; $\displaystyle \pi\approx3.14159$.

• $\displaystyle R$ – the radius; the diameter is $\displaystyle 2R$.
• A calculator has a $\displaystyle \pi$ key. Use it, and round only at the end.

A sector is the part of a disc between two radii and the arc between them. A central angle of $\displaystyle \alpha$ degrees is $\displaystyle \frac{\alpha}{360}$ of a full turn, so its arc and its sector are that part of the circumference and of the area.

Example: $\displaystyle {R}={6}$, $\displaystyle {\alpha}={60^{\circ}}$ – a sixth of the circle.

The diameter of a bicycle wheel is $\displaystyle 70$ cm.
Round to two decimal places.

A disc of radius $\displaystyle 10$ cm has a sector with a central angle of $\displaystyle 72^{\circ}$.
Round to two decimal places.

The area of a disc is $\displaystyle 50$ cm².
What is its radius, in cm? Round to two decimal places.

$\displaystyle \pi$ can be estimated with the area by the sine: inscribe a regular polygon with $\displaystyle n$ sides in a circle of radius $\displaystyle 1$, and join the centre to the vertices.

• That makes $\displaystyle n$ isosceles triangles. Each has two sides of $\displaystyle 1$, with the angle $\displaystyle \frac{360^{\circ}}{n}$ between them.
• The polygon’s area: $\displaystyle n\cdot\frac{1}{2}\cdot1\cdot1\cdot\sin\frac{360^{\circ}}{n}$.
• As $\displaystyle n$ grows, the polygon fills the disc, and its area comes closer to the disc’s – to $\displaystyle {\pi\cdot1^{2}}={\pi}$.

In the drawing, a regular octagon ($\displaystyle {n}={8}$): $\displaystyle {8\cdot\frac{1}{2}\cdot\sin45^{\circ}}={2\sqrt{2}\approx2.83}$.

The two constants – the circumference over the diameter, and the area over the square of the radius – are the same number: the height of each triangle is close to the radius, so the polygon’s area is close to half its perimeter times $\displaystyle R$. For the disc: $\displaystyle {A}={\frac{1}{2}\cdot2\pi R\cdot R}={\pi R^{2}}$.

A regular polygon is inscribed in a circle of radius $\displaystyle 1$.
Find the polygon’s area, to four decimal places.

People worked $\displaystyle \pi$ out thousands of years ago, each time a little more exactly:

• Babylon, more than 3,600 years ago: a clay tablet from Susa gives $\displaystyle {3\frac{1}{8}}={3.125}$.
• Egypt, the Rhind papyrus (about 1650 BCE): a disc of diameter $\displaystyle 9$ has the area of a square of side $\displaystyle 8$ – that is, $\displaystyle \pi\approx\frac{256}{81}\approx3.16$.
• The Bible, 1 Kings 7:23: the “sea” Solomon built was ten cubits “from one brim to the other”, and “a line of thirty cubits did compass it round about”: $\displaystyle \pi\approx3$.
• Archimedes, the 3rd century BCE: with regular polygons of $\displaystyle 96$ sides, inscribed and circumscribed, he showed that $\displaystyle 3\frac{10}{71}<\pi<3\frac{1}{7}$.
• Liu Hui, China, the 3rd century CE: with polygons of $\displaystyle 3072$ sides, $\displaystyle \pi\approx3.1416$.
• Zu Chongzhi, China, the 5th century: $\displaystyle 3.1415926<\pi<3.1415927$, and the fraction $\displaystyle \frac{355}{113}$.

Today computers know trillions of digits of $\displaystyle \pi$. It is not a fraction, and its digits after the point never end and never repeat.