Here is the derivation of the altitude equation which can be used to determine the height of a kite, or a model rocket in flight. A graphical version of this page is also available. In this derivation, * denotes multiplication, / denotes division, tan and cos are the trigonometric tangent and cosine functions.

The procedure requires an observer in addition to the kite flyer, and a tool to measure angles. The observer is stationed some distance (d) from the flyer along a reference line. To determine the altitude, the flyer must measure the angle (c) between the reference line and the location of the kite. This measurement is taken parallel to the ground. The observer must measure two angles; the angle (a) from the ground to the kite and the angle (b), parallel to the ground, between the kite and the reference line.

The height of the kite shall be called "h". We will need several "construction" triangles to derive the final equation. We will denote these triangles by the measured angle included in it. Right triangle A will be oriented vertical with a side x along the ground from the observer to the base of the kite, a side h which is the height of the kite and the angle a between side x and the hypotenuse. This gives us our first equation:

Equation 1: h = x * tan a

The second construction triangle is right triangle B. It is oriented parallel to the ground and includes side x from triangle A as its hypotenuse, side y along the reference line, and side z perpendicular to the reference line from the reference line to the base of the kite. Angle b is between side y and side x. This gives us two more equations:

Equation 2: y = x * cos b

Equation 3: z = y * tan b

The third construction triangle is right triangle C. It is oriented parallel to the ground and includes side z from triangle B and side w along the reference line and perpendicular to side z. Angle c is between side w and the hypotenuse. This gives us the following relations:

Equation 4: z = w * tan c

Equation 5: w = d - y

Combine Equation 1 and 2 by eliminating x:

Equation 6: h = y * tan a / cos b

Now let's find an expression for y. Combine Equation 3 and 4 by eliminating z

Equation 7: y * tan b = w * tan c

Combine Equation 5 and Equation 7:

Equation 8: y * tan b = d * tan c - y * tan c

Equation 9: y * tan b + y * tan c = d * tan c

Equation 10: y * (tan b + tan c) = d * tan c

Equation 11: y = (d * tan c) / (tan b + tan c)

Substitute into Equation 6:

Equation 12: h = (d * tan c * tan a) / (cos b * (tan b + tan c))

This is an example of how engineers use the math that you are studying in high school. Using a couple of simple measurements on the ground you can determine the height of a kite, or a model rocket ... which would be very hard to measure directly.


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by Tom Benson
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Responsible Official: Kathy.Zona@grc.nasa.gov
Last Updated Tue, May 05 07:13:13 PM EDT 2015 by Tom Benson