# Introduction

How can we measure the stars if we can’t reach out and touch them?

In the formative years of astronomy there was a burning desire to measure the size and distance to objects in our solar system. This tutorial will explore the following 5 questions using nothing but naked eye astronomy, geometry and trigonometry:

**How big is the earth?****How big is the moon?****How far away is the moon?****How big is the sun?****How far away is the sun?**

Let's warm up with a simple yet powerful question: how can you prove the **earth is not flat using shadows?**

# Is the earth flat?

A flat earth was once a commonly held belief, it persisted in some areas until the middle ages. Definitive proof of a spherical earth did not come until Ferdinand Magellan successfully traveled around the earth (1519-1521). However this belief was challenged much earlier thanks to major advancements in ancient Greek astronomy.

Plato (427–347 BC) was convinced that the earth was spherical: *"My conviction is that the earth is a round body in the centre of the heavens...Also I believe the Earth is very vast"*. Significantly, he could not **prove** this was the case. It was his prized student Aristotle who began offering evidence for a spherical earth. One of his more convincing pieces of evidence was the shape of the earth’s shadow visible during a **Lunar Eclipse. **Notice the subtle curvature of the shadow cast on the moon:

Another way to prove that the earth is spherical came from Eratosthenes (276–194 BC). His method involves looking at the shadows cast by the sun on the surface of the earth. This is the basis for a sundial:

But what can shadows tell us about the earth? Eratosthenes did an interesting experiment using two vertical poles located in two distant cities in Egypt. You could do this using telephone poles separated by several hundred kilometers or more. (phone a friend!)

First wait until the sun is directly overhead point A, which is the moment when the **pole casts no shadow**. At the same time have a friend observe an identical pole at point B. You will find out that the pole at point B **will cast a shadow**. Since we can assume the sun's rays reach the earth in parallel lines this is visual proof that the surface must be curved!

Try out the interactive illustration below. You can click and drag the pole to see how it affects the shadow it casts.

Pause now and think about how these shadows could allow you to determine the **radius of the earth**. We have a simulation next which will help. You will need to understand how to calculate __arc length__ so let's review that too. Have fun!

## Share a tip

When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...## Thank the author

This is great, I finally understand quadratic functions!## Have something that's not a tip or thanks about this content?

## This discussion area is not meant for answering homework questions.