The amount of the earth that is visible to an observer depends on its altitude above the surface of the earth. The higher the observer, the more of earths surface is visible. So, although the earth may always appear as a sphere from space at a certain distance, the effective amount of the surface we can see only approaches 50% for very large distances.

 The True Face of the Earth, Camera Distance matters

In this article I derive the equations in detail to calculate this amount. To understand the derivation of the area of a spherical cap some basic Calculis is required.

Equation

This equation treats the earth as a perfect sphere, which is accurate enough.

Calculation of the Cap Area of a Sphere

Principle: We define the rectangular area of an infinitely small section of the surface of a sphere. This section can be defined from two angles at the center of the sphere and the radius of the sphere. The angles correspond to the difference in latitude and longitude in a spherical coordinate system. Then we sum all such defined areas of a cap of the sphere by double integration. The smaller the areas the better the sum approximates the true area of the cap. That's why Calculus is used to get the correct area of the cap.

Area of an infinitesimal rectangle of the surface a a sphere:

Here is where Calculis comes in: To get the area of the cap we have to sum (integrate) all small rectangular areas all around the horizontal axis of the sphere (from $\varphi = 0$ to $\varphi = 2\pi$) and from the top of the cap ($\theta = 0$) to the angle $\theta$:

The inner integral (blue) correspond to the circumference of a circle on the cap around the vertical axes at the angle $\theta$. Solving the inner integral is easy. Because $\theta$ does not depend on $\varphi$ and R is constant it can be taken out of the inner integral:

The remaining outer integral to solve, after taking all constants out of the integral, is:

Solving the integral results in (integral of sin is -cos):

cos(0) = 1, so we get the area of the speherical cap:

Findings of Archimedes

Archimedes could show that the area of the cap is proportional to the height of the cap, no matter how big the sphere is. [1] Let's derive this.

The height of the cap can be calculated using trigonometry from the radius of the sphere and the angle $\theta$ as follows:

Comparing this with equation (9) we can see that part of it can be replaced by the height s:

So using the hight of the cap instead of the angle $\theta$ we get:

Note: this is the same equation as for the outer area of a cylinder with radius R and height s. This is remarkable, because it means that the cap area is the same as if we project all small rectangles of the cap to the surface of a cylinder with radius R and then sum all this projected areas.

References