Height of elevation required to see ~45 miles across Lake Huron, from Michigan to Canada?

Question

How high of a tower would have to be built to enable someone to have line-of-sight across Lake Huron from White Rock, Michigan to Canada (approximately in the area of Goderich)? According to MSN Maps, this is a distance of approximately 45 miles, or so.

Is there a formula or some calculation to approximate this, given the curvature of the earth?

Answer 1

If we include atmospheric refraction -- and we should! -- the rule of thumb is:

distance to horizon (miles) = sqrt [ 7 × h (feet) / 4 ]

So for a distance of 45 miles, the height would be 1157 feet.

Answer 2

Assumption 1: The tower you build will be at right angles to the chord connecting the base of the tower and the point on the far shore.
Assumption 2: You're looking for the "beach" with your line-of-sight.

Earth radius = 3963.1 miles.
Arc length (across Lake Huron) = 45 miles.

(Chord) = 2 * (Radius) * sin (Arc / (2 * Radius))
= 2 * 3963.1 * sin (45 / (2 * 3963.1))
= 44.9998 miles ... note that the angle calculated was measured in radians!

Since it is known that the "landing point" of your line-of-cight will be tangent with the earth's surface (minimum clearance above curvature), we can can figure out the angle of the triangle including the tower height and chord distance (on the shore side).

Sum of angles of triangle = 3.14159 radians. Anything that covers an arc is, by definition, an isoceles triangle. We already have figured that the interal (earth center) angle is 0.0057 radians. Therefore each angle at the chord is (3.14159 - 0.0057 ) /2 = 1.5679 radians.

We also know that the corner above the surface will be (3.14159 / 2) - 1.5679 = 0.0029 radians.

(Tower height) = (Chord) * tan (surface angle)
= (44.9998) * tan (.0029)
= 0.1305 miles = 689.0389 feet ... basically 689 feet.

Answer 3

About 1500 feet tall! Pretty tall!

Note: It also depends on what you want to see on the other side. This is essentially looking at the bottom of the beach. If you wish to see a house or building, you could reduce quite a bit.