Measuring the Universe

# Orbital mechanics 1

Early models of the universe were based on the assumption that circles were perfect models for the orbit of planets.

The solar system according to Copernicus. Image Credit: Harmonia Macrocosmica, Cellarius, Amsterdam, 1660

# Anatomy of a circle

Circles are symmetric in every direction, and all points on a circle have the same distance from its centre – this distance is called the radius. The distance around the perimeter of a circle is called the circumference.

Image Credit: Philliip Legner

The most fascinating feature of circles is based on the ratio between the circumference and diameter. What do these circles have in common?

What common feature do these circles share?

The ratio between the circumference and the diameter is the same for circles of any size.

Ratio of circumference to diameter is always the same. Image: John Reid

It is a mysterious number which is represented using the Greek letter  π, but what does π (pi) equal? To find out, look for a circle and divide the circumference by the diameter. You will find it is always around 3. Old Babylonian mathematics contained some estimates for  π. Tablets show they assumed π  = 25/8 which is 3.125.

Can we do better?

# How to estimate π

If you draw a polygon inside a circle, you would expect the perimeter of the polygon to be approximately equal to the circumference of the circle. This idea was developed by Archimedes (287 BC –  212 BC) who used a 96 side polygon to improve the estimate for pi. To do so he compared the perimeter of the outer polygon and inner polygon. The circumference of the circle must be between these two values.

Below is an intereactive illustration. Use the slider to change the number of sides and notice how this affects the estimate of π

As you increase the number of sides your estimate becomes more and more accurate. This leads to the following estimate for π (3.1408 < π < 3.1429) or approx 3.14

Using more sides will result in a more precise estimation of π.

An even better estimation which is handy to memorize: