In advance of a recent lunar eclipse, Fred Espenak (Mr. Eclipse) shared an image which I clip below:
In this image, it appears that the Moon's diameter is roughly equal to the width of the Earth's penumbra. Is this right?
In order to think about this, I made this GeoGebra figure:
Because it is impossible to make a scale drawing, I decided to base the figure on the apparent relative size of the various objects. During a lunar eclipse, the Earth (blue) is between the Sun (orange) and the Moon (green). To represent the fact that the latter two have approximately the same apparent size as seen from Earth, I drew sight lines from the Earth's center tangent to the Sun. If the Moon is tangent to those, it has the same apparent size as the Sun.
If you grab the Moon and move it along its orbit, you'll see that yes, the width of the penumbra there is indeed approximately equal to the diameter of the Moon. Mr. Eclipse was right! This is not as surprising as one might think: the lines that mark off the edges of the penumbra are sight lines to the top and bottom of the Sun. Therefore any object with the same apparent size as the Sun will have a diameter equal to the width of the penumbra at that distance. As it turns out, that is the case (more or less) with the Moon.
If you want to play around with the figure, click on the controls checkbox. You can change the various radii by using the control segments that appear.
You can get the figure back to its original state by clicking the button at the top right.
Thanks to Bruce Cohen for his help in building this!
Download the GeoGebra figure.