What Is This Site All About?

This site was developed as a project in a primary school (aged 9 and 10) science fair. As a father and daughter team we set about answering the following questions :-

• If you dug a tunnel straight down, where would you come out? This location on the other side of the globe is called the 'antipode'.
• The phrase 'as the crow flies' suggests the shortest route between two points, but is this true when the world is a sphere?
• Can we create a tool that can easily show the distance between two locations anywhere in the world, by going through the Earth instead of going over its surface?

How Do You Use This Site?

• The tunnel start location and destination location can be dragged around the map and the distance calculators will automatically update.
• The red line shows the tunnel, and the purple line shows the distance travelled around the globe.
• You can also enter a start and destination location using the search input boxes. This can be a co-ordinate, postcode, address, city, or country. A Google search will attempt to find the location that best matches your input.
• If you click on the 'Antipode of Start' button the destination location will be moved to the opposite side of the world to the start point.
• Clicking on the location icons will provide a pop-up with details about the location.
• The elevations button will produce a graph showing elevation above (and below) sea level if the journey was taken across the surface of the world. This means walking up mountains and under the water. It breaks the journey up into 256 points, so the longer the journey the less accurate it will be.
• Clicking on the button will load the map in full screen. Clicking on the button will bring back this page.
• All the usual Google Map functions are available, so you can switch between satellite and map view, and you can zoom in and out, scroll around etc.

How The Maths Works

First we need to simplify the idea of working with a sphere. Imagine lining up the two points on a globe with the centre of the globe and then slicing it in two along that line (much like cutting a water melon in half).

You now have a circle that still has the same radius as the sphere, and the two points are on the edge of the circle.

Maths with circles is a lot simpler than spheres, so without crunching any numbers we now have a simpler equation to solve.

In this example we are going to calculate the tunnel distance between London and New York. Google Maps already provides us with the flying distance between locations. In maths the distance between two points on the circumference of a circle is called the 'arc'. The length of the arc between these two locations is 5567 kilometres (km).

We also know that the 'radius' of the Earth is 6371 km. That is the distance from the centre of the Earth to the surface - obviously the Earth is a lumpy old thing so this value is an average.

We calculate the radian 'X' by dividing the 'arc' by the 'radius'. So in this example X = 5567 ÷ 6371 which give a value of 0.8738.

Next we calculate the radian 'Y' by subtracting 'X' from π (approximately 3.1416) and dividing that value by 2. In this example Y = (3.1416 - 0.8738) ÷ 2 which give a value of 1.1339;

Note 1: As the radius is the same for both sides of the triangle we know we have an isosceles triangle. That is, two of the corners are the same angle. This means 'Y' is the same on both corners. Dividing the value by two gives us the value of each single 'Y' angle.

Note 2: A radian of 1 is equal to 57.296 degrees. And as we know, the sum of all the angles of a triangle equal to 180 degrees. If we add the three radian values together we get 3.1416 (which is π), and if we multiply π by 57.296 we get 180. Cool hay!

The final step is to calculate the distance between two points on the arc. This line is called the 'chord', and is the route our tunnel will take. The equation is completed using the SIN function. This is one of the buttons on your calculator that you never usually use.

Radius × sin(X) ÷ sin(Y). In our example, the tunnel distance is 6371 × sin(0.8738) ÷ sin(1.1339). This gives us the answer of 5392 km.

Frequently Asked Questions

Q.	Is the world a perfect sphere?
A.	No. It is an oblate spheroid. That means it's squashed at the top and bottom, and rounder in the middle. At the equator the world is 21km (13 miles) wider than measuring from pole to pole. The main reason for this is down to the earth spinning. The similar effect can be felt when you spin yourself and put out your arms, you can feel your arms being pulled outwards.
	For the benefit of simplicity Google Maps treats the world as a sphere and uses the average radius of 6731km.
Q.	Where does this tool get its data from?
A.	The tool uses the Google Map Javascript API software. All location searches, co-ordinates, and elevation data comes from Google.
Q.	Can we contact you to provide feedback, report problems, or request enhancements?
A.	Yes, we'd love to know if you found the tool fun and useful. Feel free to contact us here with your emails.
Q.	The map won't load, has something broken?
A.	Google provide the fabulous underlying software used in this site. They limit the number of uses to 25,000 map loads every day. Should the map fail to load it is most likely because this limit has been reached. If you try again later it should work again.

Conclusions

Q.	If you dug a tunnel straight down, where would you come out? This location on the other side of the globe is called the 'antipode'.
A.	If you dug a tunnel straight down it would always come out the other side of the world. However if you dug a tunnel it would most likely come out in the ocean. The world's surface is made up from 29% land and 71% water. Of that land mass 24.6% has an antipode that ends in the ocean, and 4.4% has an antipode that ends on dry land.
Q.	The phrase 'as the crow flies' suggests the shortest route between two points, but is this true when the world is a sphere?
A.	No, it would be the second shortest route as the crow would go around the circumference whilst an antipode tunnel would use the diameter. 20037.51km is the semi-circumference and 12741.98km is the diameter. The tunnel distance is 7295.53km shorter then having to go all the way around the circumference of the world. This is a journey saving of 36.4%.
Q.	Can we create a tool that can easily show the distance between two locations anywhere in the world, by going through the Earth instead of going over its surface?
A.	We hope this website shows that we can do exactly that. We hope it helps you find out more about our world, and you enjoy finding the antipode of where you live (even if it ends with you getting wet feet).