Horizon logic problem?

Question

You are standing in the middle of a flat desert, admiring the distant sky. What a lovely moment, until your mind wanders and you ask yourself: How far away is the horizon?

Well, how far away is it?

*Please go through your thought process and exactly what you did to get the answer.

Answer 1

Draw a circle centre O to represent a cross section through the earth.

Draw a vertical radius OP meeting the cirlce at P, and extend it to Q so that PQ equals the height of your eyes above the ground at P.

From Q draw a tangent to meet the circle at R.

Then angle QRO is a right angle (tangent / chord).

Let:

r be the radius of the earth,

h be the height of your eyes above ground,

s be the distance of the horizon.

The square on the tangent equals the rectangle contained by the secant OQ and that part PQ of it which lies outside the circle.

QP . QO = QR^2

h(h + r) = s^2

Using r =  6.378 * 10^6 m and h = 1.67 m:

s = sqrt[ 1.67 * 6.378 * 10^6 ]

= 3264 m

= 3.26 km.