Let R be the radius of the earth. It doesn't matter what R is. This R value can change (say we go to another planet, the idea still holds). In the drawing, I made R = 2, but again the value for R doesn't matter.
Place point A at the origin (0,0). Place point B to be R units away from point A. So let's say B = (R, 0). Draw a circle centered at point A and that goes through point B. The equation for this circle is x^2+y^2 = R^2. Call this circle p.
Place point C at (0,R). Draw a ray from point A that extends through point C and goes on forever from there. Plot a point D such that D is on the ray but not between A and C. This point D is going to represent the person's location. Since the ray is a vertical line, we only need to worry about the y coordinate of point D. If the person's height off the ground of the planet is k, then D = (0,k+R).
We have
A = (0,0) and D = (0,k+R)
Find the midpoint of A and D to get E = (0,(k+R)/2)
Draw a circle centered at point E and that goes through point A, or point D. This circle will have the equation x^2 + (y - (k+R)/2)^2 = ((k+R)/2)^2. Call this circle q
The two circle equations are
p: x^2+y^2 = R^2
q: x^2 + (y - (k+R)/2)^2 = ((k+R)/2)^2
Subtract the equations p-q to get
[x^2+y^2] - [x^2 + (y - (k+R)/2)^2] = R^2 - ((k+R)/2)^2
y^2 - (y - (k+R)/2)^2 = R^2 - ((k+R)/2)^2
y^2 - (y^2 - 2*y*(k+R)/2 + ((k+R)/2)^2) = R^2 - ((k+R)/2)^2
y^2 - (y^2 - y*(k+R) + ((k+R)/2)^2) = R^2 - ((k+R)/2)^2
y^2 - y^2 + y*(k+R) - ((k+R)/2)^2 = R^2 - ((k+R)/2)^2
y*(k+R) = R^2
y = (R^2)/(k+R)
What does this mean? Well it means that circle p and circle q cross at two points, call them F and G. Point F and point G have the same y coordinate, and that y coordinate is equal to (R^2)/(k+R)
The vertical distance from point F to point C is
R - (R^2)/(k+R) = (R*(k+R))/(k+R) - (R^2)/(k+R)
R - (R^2)/(k+R) = (R*k+R^2)/(k+R) - (R^2)/(k+R)
R - (R^2)/(k+R) = (R*k+R^2-R^2)/(k+R)
R - (R^2)/(k+R) = (R*k)/(k+R)
R - (R^2)/(k+R) = (k*R)/(k+R)
This is the height h of the spherical cap as shown on this article. Scroll to the bottom of that page and you'll see the formula
Let's plug in
If we let k = x, then we can make the function
We're told that we want the surface area of the cap to be 1/4 of the surface area of the planet (aka sphere). So,
We want the surface area of the cap to be
Plug this into S(x) and solve for x
It turns out that if the height of the person is exactly equal to the radius of the planet, then the person will be able to see 1/4 of the sphere's surface area
The radius of the Earth is roughly R = 3959 miles
So the person should be at a height of x = R = 3959 miles off the ground
Note: The earth isn't a perfect sphere, but I'm assuming it is just to make things a bit more simple.