Subject: arc
Name: Thomas
Who are you: Student

what is the height of the arc of the earth over a certain distance?

Hi Thomas,

I drew an arc of a circle and labeled some points and distances.

In my diagram r is the radius of the Earth, the distance from D to A is 2x, B is the midpoint of the line section DA and h is the unknown height. I haven't included units, you can use whatever units you wish as long as you use the same units for each of the distances.

The triangle ABC is a right triangle and hence, by Pythagoras' Theorem

r² = x² + (r-h)²

which simplifies to

h² - 2rh + x² = 0

Solve for h.

Penny