The "Three Satellites" problem

There are three satellites orbiting the Earth over the equator. They are equally spaced around a circular orbit. In the diagram below, you are looking down on the north pole of the Earth, with the equator running around the edge of the earth's disk. In order that any point on the equator is visible by at least one satellite, their positions form an equilateral triangle.

If the satellites are in the lowest orbit possible, the Earth is the incircle of the triangle, and the orbit is the circumcircle.

Assume the earth has a circumference of 24,000 miles.

In the problems below, online users can click 'check answer' to get an approximate answer to the problem. If using a printed copy, write your answer in the box.

Problem 1
What is the radius of the Earth, to nearest mile? show answer
Somewhere between 3500 and 4000 miles
Problem 2
What is the size of the equilateral triangle linking the three spacecraft? show answer
Each side is somewhere between 13,000 and 13,500 miles long.
Problem 3
What is the radius of the orbit? show answer
Somewhere between 7500 and 7800 miles.
Problem 4
How high above the Earth's surface are the spacecraft flying? show answer
Somewhere between 3500 and 4000 miles.
Challenge problem 5
From the above results, can you see a connection between the answers to question 1 and 4?
What does this mean in mathematical terms?

Resources (and perhaps clues)