The Moon has 27.3% of the Earth's radius/diameter but only 2% of the Earth's volume. I don't quite get it why the Moon's volume is that small despite having more than a quarter the 2-dimensional size of Earth. Compared to the Earth, does the Moon really have 2% of the Earth's 3-dimensional size? Why is it like this with the Moon and other bodies of similar diameters?
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2$\begingroup$ That's just geometry. $(0.273)^{3} \approx 0.02035$ $\endgroup$– PM 2RingJul 17, 2020 at 5:40
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$\begingroup$ Look the volume of a sphere (close enough for comparison). $\endgroup$– FredJul 17, 2020 at 5:42
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$\begingroup$ @PM2Ring From google I get a result of 0.085. That would sound credible for the Moon. $\endgroup$– IoannesJul 17, 2020 at 5:57
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$\begingroup$ I dont know what you mean. Where does that 0.085 come from? BTW, the radius is a 1 dimensional measure of size, not 2 dimensional. From Google, I get 1,737.1 km & 6,371 km for the radii of the Moon & the Earth. Their ratio is ~= 0.2727, (which is slightly smaller than the value given in your question), so the ratio of their volumes is the cube of that, ~= 0.02027 $\endgroup$– PM 2RingJul 17, 2020 at 6:26
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4$\begingroup$ But you don't need to worry about the formula for the volume of a sphere. If you have two 3D objects of identical shape, and the ratio of their linear sizes is $k$, then the ratio of their volumes is $k^3$. $\endgroup$– PM 2RingJul 17, 2020 at 8:59
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2 Answers
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...despite having more than a quarter the 2-dimensional size of Earth.
I think herein lies the problem; diameter is a 1-dimensional measurement, it's units are distance.
Let's rewrite 27.3% as 0.273. If that's the ratio of diameters, then the ratio of 2-dimensional areas should be (0.273)2 and the ratio of the volumes should be (0.273)3. Those numbers are 0.0745 and 0.0203 respectively.
So the next question is why with a volume of 2% of Earth that The Moon's mass is only 1.2% of Earth's!? That's because the Moon's average density is only 3.3 g/cm^3 compared to 5.5 g/cm^3 for Earth.
For more on that see answer(s) to Are there any known asteroids with average density similar to that of Earth's?
Visual cues are deceiving, these may help. Note that the larger beaker with the red fluid is filled to 600 ml, and the smaller one with blue fluid is only 60 ml, it looks like it's double in size but it's roughly 10x larger in volume!
Sources: above: https://en.wikipedia.org/wiki/File:Beakers.jpg and below: Why does the Moon appear gray when passing between the Sun and the Earth?
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$\begingroup$ @Greenhorn it's not intuitive, our brains don't naturally think that way I know. But it's really true! If you can find small and large drinking glasses or glass beakers of the same shape but different sizes, try pouring one into the other. If the large one is twice the size of the small one, you'll have to fill and empty the smaller one eight times to fill the larger one. 60 ml vs 600 ml: en.wikipedia.org/wiki/File:Beakers.jpg $\endgroup$– uhohJul 17, 2020 at 8:19
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3$\begingroup$ This might be entirely off-topic, but isn't the larger beaker filled with a pale greenish yellow liquid and the smaller beaker filled with a red liquid? Or are my eyes playing tricks on me?! $\endgroup$– user26067Jul 17, 2020 at 12:58
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$\begingroup$ My eyes are also fooling me, the stuff in the large beaker looks pale green and the stuff in the small beaker looks bright red. $\endgroup$ Jul 17, 2020 at 14:04
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It is all just geometry and mathematics.
The volume of a sphere is calculated according to this formula:
Volume = $(4/3) \times \pi \times r^3$
where $\pi$ = 3.14159..., and $r$ is the radius of the sphere.
The Earth radius is 6,371 km.
The Moon radius is 1,737 km.
We put the numbers into the formula and we get:
• The volume of Earth is 1,083,206,916,845.7535 km³ (one trillion eighty-three billion bla-bla-bla cubic kilometers)
• The volume of Moon is 21,952,706,175.030006 km³ (twenty-one billion nine hundred and fifty-two million bla-bla-bla cubic kilometers, which is approximately twenty-two billion cubic kilometers)
Since one trillion is one thousand billion, you don't even need a calculator to understand that one trillion is roughly 50 times bigger than 21 billion, and one fiftieth (1/50) is exactly 2%.
Here is a picture of Earth and Moon shown as balls lying close to each other. You can use a ruler and see that the picture is true to life, the diameter of the smaller ball is really 27.3% (about 1/4) of the diameter of the bigger one:
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$\begingroup$ My question is concerning the volume despite the diameter, not the diameter despite the volume. But thank you nonetheless. $\endgroup$– IoannesJul 17, 2020 at 9:08
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3$\begingroup$ @Greenhorn - The volume of a sphere depends only on its diameter/radius, nothing else. $\endgroup$ Jul 17, 2020 at 9:58
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$\begingroup$ I know. You seem to make the assumption that it's the diameter I'm wondering about. I'm wondering about the volume, not the 1-dimensional size. $\endgroup$– IoannesJul 17, 2020 at 10:44