Earth Radius Calculator

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Calculating the radius

We can also use the calculator to find the radius given the observer height, the distance to the object, the Coefficient of refraction and the hidden height. We will look at an example of this on the next page.

On the calculator, you need to fill in the top part, then go to the bottom part to fill in the hidden height.

To the calculator

The derivation of the formulas

From the previous page, we have

$\displaystyle d=\sqrt{{2Rh+{{h}^{2}}}}$

Similarly we get

$\displaystyle D=\sqrt{{2RH+{{H}^{2}}}}$

From L=d+D we get

$\displaystyle L=\sqrt{{2Rh+{{h}^{2}}}}+\sqrt{{2RH+{{H}^{2}}}}$

Squaring this, we get

$\displaystyle {{L}^{2}}=2Rh+{{h}^{2}}+2\sqrt{{2Rh+{{h}^{2}}}}\sqrt{{2RH+{{H}^{2}}}}+2RH+{{H}^{2}}$

$\displaystyle {{L}^{2}}-{{h}^{2}}-{{H}^{2}}-2R(h+H)=2\sqrt{{2Rh+{{h}^{2}}}}\sqrt{{2RH+{{H}^{2}}}}$

Let us set

$\displaystyle Q={{L}^{2}}-{{h}^{2}}-{{H}^{2}}$

This gives us

$\displaystyle Q-2R(h+H)=2\sqrt{{2Rh+{{h}^{2}}}}\sqrt{{2RH+{{H}^{2}}}}$

Next, we square and divide through by four to get

$\displaystyle \frac{{{{Q}^{2}}}}{4}-R(h+H)Q+{{R}^{2}}{{(h+H)}^{2}}=(2Rh+{{h}^{2}})(2RH+{{H}^{2}})$

Say we want to write this as

$\displaystyle a{{R}^{2}}+bR+c=0$

After some algebraic manipulation, we get

$\displaystyle \left\{ \begin{array}{l}a={{h}^{2}}-2hH+{{H}^{2}}\\b=-(h+H)Q-2h{{H}^{2}}-2{{h}^{2}}H\\c=\frac{{{{Q}^{2}}}}{4}-{{h}^{2}}{{H}^{2}}\end{array} \right.$

We can now solve the equation. Our solution is

$\displaystyle R=\frac{{-b-\sqrt{{{{b}^{2}}-4ac}}}}{{2a}}$

If h=H we will get a=0, and thus not a quadratic equation. If a<10^-13, we instead solve the equation

$\displaystyle bR+c=0$

The radius R in the above equations is the apparent radius. To get the actual radius, we calculate

$\displaystyle {{R}_{{calulated}}}=R\cdot (1-k)$

Finally, we calculate the value k would have to have to give us the exact answer.

Up a level : The Shape of the Earth
Previous page : Earth Curvature Calculator
Next page : Radius calculations - an example

Last modified: May 4, 2025 @ 11:24