# How everybody can measure the size of the Earth

It may seem that measuring the Earth is a feat best left to the experts with modern technology. Or it may seem that measurement on such a grand scale was a great feat of the ancients far removed from our day to day world. But neither of these is true. We all live on the same planet and there really isn’t anything stopping you from doing it yourself.

Materials: A stick or other object to cast a shadow, protractor (or tape measure if you want to use the trigonometry method of finding the angle), string, a compass (optional), Plumb bob (optional), a mode of traveling more than 120km (75mi) or more to the north or south (I used a car) and a way of measuring the distance traveled (I used my odometer in my car).

Experimental steps:

1. While the first few steps can be done in any order it’s usually a good idea to decide on a set of cities to compare first. Ideally they will be  nearly on the same line of latitude. Also they need to be farther apart than 120km (75mi) because anything less than this and you will have less than a degree of change between the locations and you won’t have a good values.
2. Next plan on the timing of the measurement. It ought to be done at solar noon. If it can’t be done exactly then be sure you use the same time in both measurements.
3. At your first location arrange your stick so that it stands at 90° to the ground. If this is difficult, use a plumb bob hung from the stick to see where vertical is. Also check the direction of the shadow. In the Northern hemisphere it should point North and in the southern South. Beware the angle of magnetic declination. The shadow will point to true North but magnetic North is slightly off from this. Also if it is too far from solar noon the measurement will be invalid.
4. At the appointed time pin down the string at the tip of the shadow (bright day is best) then holding this in place streatch the string up to the tip of the stick (an assistant can be helpful here). Hold the protractor so that is is at a good right angle with the ground and measure the angle of the string. Alternately measure the length of the shadow along the ground and the height of the stick.
5. Travel to the second location withing a few days and make the same measurements there. Note the measurement closer to the equator should have a smaller angle as the shadow will be shorter. If not chances are the times the measurement were taken are not the same or you did not travel far enough.
6. If you chose to measure the shadow instead of the angle the angle can be found by taking tan-1(base/height). This method is also more accurate than the one shown in the video.
7. Finally refer to the distance between your two locations and divide this by the the difference of the angle: distance/(angle 1-angle 2) then multiply this value by 360degrees to get the final value.

Data:
I took multiple measurements at each location and averaged them to get the final measured angle.
First location(57°+59°+59.5°)/3=58.5°
Second Location (59°+60°+61°)/3=60 °

Math:
108mi/(60°-58.5°)=108/1.5°

108mi/1.5°*360°=25920mi or 41700km

Error analysis: I have to say measuring the angle directly was very sloppy. In hind sight I should have used the pathogen method to find the angle with the ground, but I wanted to avoid trigonometry with this video. Alternately I could have gone farther south to perform the same task. Likewise my accuracy plus or minus could have been improved. I had only about one quarter degree accuracy at best but that gives me wildly different values for the earth. Still I found that it fit well with the prediction. Comparing the final value I got to the accepted value of the Earth’s equator I was about 1000 miles (1700km) too large.

This leaves a relative error of:
(25920-24901)/24901*100%=4.0%

Although that’s the value for the equatorial circumference. In truth the Earth is oblate because of the centripetal acceleration at the equator. The north south circumference is 24859mi which gives:

(25920-24901)/24901*100%=4.2%

Still this is as good as Eratosthenes had gotten for the size of the Earth and well within an order of magnitude of the true value.

Discussion:
Usually the measurement of the earth is discussed by talking about Eratosthenes but despite the simplicity and applicability of his method very rarely does anyone do the measurement. It’s left in a very abstract state with out of scale pictures and only a vague discussion of how the seasons effect the measurement. Also it usually gets pretty heavy into the trigonometry since in ancient times the distance between two cities was one of the last things Eratosthenes figured out.

Hopefully I’ve demystified this classic experiment and given you he tools to try it yourself.