Planetary Density Calculator

ρ = 3M / 4πR³ — bulk density is the single number that constrains what a planet is made of, without ever landing on it.

Starting values — Planetary Density Calculator

Mass (M)	5.972 × 10²⁴
Radius (R)	6.371 × 10⁶

Change any control above and the results recompute from these.

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Lesson

The theory — Planetary Density Calculator

Mean density is a planet’s total mass divided by its total volume — one number for the whole body. It is not the density of anything you could pick up: it is the average over crust, mantle and core together.

What each symbol means

ρ: the mean density, shown here in grams per cubic centimetre so the numbers are comparable to familiar materials — water is 1, iron about 7.9.
R: the planet’s radius, 6.371 × 10⁶ m for Earth. It enters cubed, so a 10% error in radius moves the density by about 30%.
3/4π: the geometric constant that appears once you assume a sphere — nothing physical, just the shape.

Where the formula comes from

Density is mass over volume: ρ = M / V. Nothing about planets yet.
Assume the body is a sphere, whose volume is V = (4/3)πR³.
Substitute and tidy: ρ = M / ((4/3)πR³) = 3M / (4πR³). That is the expression on the page, and at Earth’s mass and radius it gives 5.513 g/cm³.

Assumes: A sphere, and that mass and radius are both known. Neither is measured directly for a distant planet: mass comes from its gravitational pull on something else, radius from how much starlight it blocks — so a bulk density inherits the error bars of both.
Breaks when: A mean hides structure completely, and Earth is the clearest case: the mean is 5.513 g/cm³, yet its core is roughly twice that and its crust about half. Two planets built quite differently can share a mean density exactly, so the number constrains what a world is made of without ever determining it.

density as a planetary X-ray 🖖

We have never drilled more than 12 km into Earth, yet we know its core is iron. The trick is bulk density: ρ = 3M/(4πR³). Mass comes from timing orbits of moons or spacecraft; radius from occultations or radar altimetry. The resulting density, cross-referenced against equations of state for rock, ice, and iron at high pressure, narrows down what mix of materials must be inside. A density below ~2 g/cm³ rules out a significant rock fraction. Above ~5 g/cm³, a large iron core is nearly unavoidable. Seismology and moment-of-inertia measurements refine the picture — but density alone does most of the work before any probe is sent.

what packing tells you about a world 🖖

Density just measures how much material is crammed into a given space, in grams per cubic centimetre. Comparing that one number across worlds lets you guess what a planet is made of without ever landing. A striking case: Saturn's average density is only about 0.69 g/cm³, less than water, so a big enough ocean would let it float, while rocky Earth at 5.51 g/cm³ is the densest planet in the Solar System.

gravity fakes a planet's density 🖖

Large planets squeeze themselves: their own gravity compresses the interior, so the measured bulk density runs higher than the material's natural value. Astronomers correct for this to recover the uncompressed density. Do it, and Mercury, not Earth, comes out densest at roughly 5.3 g/cm³ versus Earth's ~4.4, betraying an outsized iron core that fills about 83% of its radius.

Practice

Check yourself

Predict the answer first, then use the controls above to find out. Reveal only after you have committed to a guess — that is what makes it practice.

Radius enters cubed, so a small error in it is a large error in density. Earth's 6.371 × 10⁶ m gives 5.513 g/cm³. Raise the radius 10% to 7.008 × 10⁶, then lower it 10% to 5.734 × 10⁶. Are the two errors the same size?

Show answer

No, and the difference is worth knowing before quoting an error bar. Ten percent too large gives 4.142 g/cm³, a fall of 25%. Ten percent too small gives 7.563 g/cm³, a rise of 37%. A cube is not symmetric about 1: 1.1³ = 1.331 but 0.9³ = 0.729. So the same slip in radius hurts half again as much in one direction, and a density inferred from an underestimated radius is the one to distrust.
Press Mercury, then Earth. The two means come out 5.427 and 5.513 g/cm³, within 2% of each other. Does that make them similar worlds?

Show answer

No — and the mass boxes show the catch before any correction does: Earth's 5.972 × 10²⁴ kg against Mercury's 3.301 × 10²³ is a factor of eighteen. A big planet squeezes itself, so Earth's mean overstates what its material would be at rest while Mercury, far lighter, barely compresses at all. Undo that and the two swap places, Mercury near 5.3 against Earth's 4.4. A mean density is a strong constraint and a weak conclusion: it rules out what a world cannot be made of long before it says what it is.

Problem solved in full

Surface gravity from Earth's mean density of 5.513 g/cm³ and surface rock 6 steps
Earth's mass and radius give a mean density of 5.513 g/cm³. Surface rock is about 2.9. Work out what that forces, then get surface gravity out of the same two numbers.
1. Mean density is mass over volume, and the volume of a sphere is where the radius does its work — cubed, so a 1% error in R is 3% in ρ.
  
  $ρ = \frac{M}{\frac{4}{3} π R ^{3}} = \frac{5.972 \times 1 0 ^{24}}{\frac{4}{3} π ( 6.371 \times 1 0 ^{6} ) ^{3}} = \frac{5.972 \times 1 0 ^{24}}{1.0832 \times 1 0 ^{21}} = 5513 kg/ m^{3}$
2. Convert to the unit geologists use, and put it beside a rock. Basalt is 2.9, granite 2.7, and the Earth's average is nearly double both.
  
  $ρ = 5.513 g/c m^{3} against surface basalt at 2.9 g/c m^{3}$
3. Surface gravity from the same inputs. The 9.82 is slightly above the standard 9.80665 because the standard value includes the rotation and the equatorial bulge, and this sphere has neither.
  
  $g = \frac{GM}{R ^{2}} = \frac{6.674 \times 1 0 ^{- 11} \times 5.972 \times 1 0 ^{24}}{( 6.371 \times 1 0 ^{6} ) ^{2}} = 9.82 m/ s^{2}$
4. Substitute the density back into the gravity formula and the mass disappears. Gravity at the surface of a uniform sphere is proportional to ρR and to nothing else.
  
  $g = \frac{G \cdot \frac{4}{3} π R ^{3} ρ}{R ^{2}} = \frac{4}{3} π GρR ⟹ g \propto ρR$
5. Test that on a body with a very different mass. The Moon is 1.2% of Earth's mass but 27% of its radius and 61% of its density, and the formula gets its gravity right.
  
  $Moon : ρ = 3340, R = 1.737 \times 1 0^{6} ⟹ g = \frac{4}{3} π G (3340) (1.737 \times 1 0^{6}) = 1.62 m/ s^{2}$
6. One more line gives the escape speed, which is the same energy statement written for a departing object rather than a standing one.
  
  $v_{esc} = \frac{2 GM}{R} = 2 g R = 11186 m/s$
Answer

It forces an interior about twice as dense as anything you can pick up. The mean is a volume-weighted average, and the crust is a rounding error in that volume — so if the outer part is 2.9, the inner part has to be far above 5.5 to pull the average there, and the only cosmically abundant material that dense is iron. That single arithmetic step, available to anyone with Cavendish's G and a mean radius, is why we knew the Earth had an iron core long before seismology could see it. The same two numbers give surface gravity, and the tidy form g = (4/3)πGρR shows that gravity depends on density and radius only, not mass — feed it the Moon's 3340 kg/m³ and 1738 km and it returns 1.62 m/s², which is the number Apollo walked in.

$ρ = 5.513 g/c m^{3} ≫ 2.9 ⟹ something much denser is underneath$

References (1)

Earth's mass and radius, as the nominal values adopted by the IAU: A. Prša et al., "Nominal Values for Selected Solar and Planetary Quantities: IAU 2015 Resolution B3." The Astronomical Journal 152(2), 41, 2016.

By the 42Math team · Updated 2026-07-29

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

All Astronomy & Space lessons →

Example problems

Mercury - Mercury — 5.43 g/cm³, anomalously iron-rich core (83% of radius)
Venus - Venus — 5.24 g/cm³, Earth’s near-twin in size, mass and density alike
Earth - Earth — 5.51 g/cm³, the densest planet in the Solar System
Mars - Mars — 3.92 g/cm³, rocky but with a proportionally smaller core than Earth
Jupiter - Jupiter — 1.33 g/cm³, the densest of the four giants after Neptune
Saturn - Saturn — 0.69 g/cm³, less dense than water; it would float
Uranus - Uranus — 1.27 g/cm³, the least dense of the giants apart from Saturn
Neptune - Neptune — 1.64 g/cm³, denser than Uranus despite being nearly the same size
Moon - The Moon — 3.34 g/cm³, rocky with only a very small core
Pluto - Pluto — 1.85 g/cm³, roughly two-thirds rock to one-third ice