How Far is the Horizon? - Stephanie Wong

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How Far is the Horizon?
by Stephanie Wong

The best way to solve this question is to use trigonometry! But first, we need to make some assumptions:

- The Earth is completely spherical.
- We are looking at a "flat" horizon that is unobstructed.
- There is no atmosphere to obstruct the view or bend light.

We draw a simple picture depicting the Earth (a circle) and the person (straight line) whose eyes are located at 1.8 metres above the ground. No matter where on Earth the person is located, he/she is located right above the centre of the Earth. The distance from the person's feet to the centre of the Earth is the same as the Earth's radius R=6,378,100 metres (equatorial radius). That means the distance from the person's eyes to the center of the Earth is 1.8+R.

The distance from the person's eyes to the horizon is denoted by the red line. Where this line touches the surface of the sphere is a tangent line to the sphere at that point. The distance from this point to the center of the Earth is also R. Drawing these lines gives us a right triangle:

By the Pythagorean Theorem, we can figure out the length of one side of a right triangle if we know the length of the other 2 sides. The formula is:

c² = a²+b², where c is the side opposite that of the right angle.

Invoking this formula, one can find the distance from the person's eyes to the horizon to be:

distance = sqrt((1.8+R)²-R²)

If you would rather know the walking distance to the horizon, you will need to know the angle q. To get q, one can use the Law of Sines (or the Sine Law):

a/b = sinA/sinB , where a and b are the lengths of two sides of a triangle, and A and B are their opposite angles, respectively.

We know one angle already, the right angle (90°). Its corresponding side has a length of 1.8+R. Angle q's corresponding side is of length sqrt((1.8+R)²-R²). That is all we need.

(1.8+R)/(sqrt((1.8+R)²-R²)) = sin(90°)/sin(q)

Since sin(90°)=1, and rearranging, we get:

q = sin^-1((sqrt((1.8+R)²-R²))/(1.8+R))

Now, we know that a full circle is 360°. Plus, from the formula to get the circumference of a circle,

C = 2pR ,

we see that 360°=2pR , so that the arc that subtends a degree of 1 is:

length = pR/180°

We want the length of the arc formed by q, so just multiply each side by q:

arc length = pR/180*q

This is how far you would have to walk to get to the horizon (of course, your horizon shifts as you walk!).

- 21 August 2005

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Last Updated:
21 August 2005