Speaking Volumes

A puzzle by A. Kozlov from the Soviet popular science magazine Kvant:

Watching a solar eclipse, a girl asks her father how much farther away is the sun than the moon.

He says, “As far as I remember, 387 times farther.”

She says, “Then I can figure out how much greater is the sun’s volume than the moon’s.”

He thinks about this and says, “I think maybe you can.” How can she do this?

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During a solar eclipse the moon covers the sun almost perfectly. This means that the ratio of the sun’s radius to the moon’s radius is the same as the ratio of these two bodies’ distance from Earth. So the sun’s volume is larger than the moon’s by a factor of about 387³.