# What is the diameter size of the umbra shadow cone of the Earth when the Moon passes through it on a lunar eclipse?

I am sure this varies given the distance from moon to earth varies, but a range would be sufficient. I am trying to explain to a flat earther how there is not a lunar eclipse every full moon.

My approach is to show that the 5 percent inclination of the moon's orbit to the ecliptic will result in a large enough gap for the umbra to miss the moon when projected at a great distance.

So the length of this cone is: $$\tan\left(\frac{\theta}{2}\right) = \frac{R_{earth}}{L}$$ $$L = \frac{R_{earth}}{\tan\left(\frac{\theta}{2}\right)}$$ $$L = 1.4\times 10^9 \text{m}$$ The diameter of the umbra varies linearly as your displacement from the vertex. The moon at perigee is about $$0.36\times 10^9\text{m}$$ from earth or about $$1.0\times 10^9\text{m}$$ from the vertex.
Therefore the diameter of the umbra at that distance is approximately $$\frac{1.0}{1.4} D_{earth}$$ or about $$9100\text{km}$$. Smaller than that when the moon is further away.