Calculating Horizon as a Function of Altitude ```Name: Patrick Status: student Age: N/A Location: N/A Country: N/A Date: N/A ``` Question: If I am in an airplane flying at 37,000 ft. and I look out my window, how far in the distance (land) can I see? Replies: Roughly 240 miles. To get this answer, draw a circle, representing the Earth, and define the following three points: A - where the plane is, at height h=37000 ft. above the Earth's surface, and at distance R+h from the center of the Earth. (R is the Earth's radius. R ~ 4000 mi. ~ 21,000,000 ft.) B - the horizon as seen from point A C - the center of the Earth The line AB is tangent to the circle, so it must make a right angle with the line BC. This makes ABC a right triangle, so (AB)^2 + (BC)^2 = (AC)^2 (AB) is the distance we want; let's call it 'd'. (BC) is just the Earth's radius, R. (AC) is R+h. In these terms, the equation is d^2 + R^2 = (R+h)^2 d^2 = (R+h)^2 - R^2 = h^2 + 2hR, which we can approximate as 2hR, since h is small compared to R. So d ~ sqrt(2hR) Tim Mooney Assuming the earth is a perfect sphere with radius about 3950 miles and that you can see to the horizon, you should be able to see about 16.6 miles. The idea is that the line segments joining the center of the earth to you in the plane and to the horizon along with the segment from you to the horizon form a right triangle with the segment from you to the center being the hypotenuse. Let h be your altitude, r the radius of the earth and x the distance from you to the horizon, x^2 + r^2 = (h + r)^2. Multiplying out, yields x = sqrt(h^2 + 2hr). Substituting (3950*5280ft for r and 37000ft for h ) yields x = 1242860 feet or about 235 miles, approximately. P. Beem Click here to return to the Mathematics Archives

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