Is the Earth Really Smoother Than a Billiard Ball?
― Shrink it down, and Everest is half the thickness of a hair
Mountains over 8,000 metres high and ocean trenches deeper than 10,000 metres are surprisingly small bumps next to the size of the Earth. Shrink the planet and do the sums, and it turns out to be a smooth, pleasant sphere. The sea and air we live in are a film as thin as kitchen wrap over its surface.
Look down from an airplane window and the mountain ranges rise in rugged rows. The ground looks very uneven. Climb a mountain and you feel its height in every muscle.
Yet a photo of the Earth taken from space shows a smooth blue ball. You can't see the shadows of the mountains or the dents of the ocean trenches at all.
You may have heard that "the Earth is smoother than a billiard ball." Is that just a figure of speech? Or is it true when you do the maths?
The answer hinges on two comparisons
The gap from the top of the highest mountain to the bottom of the deepest sea is about 20 kilometres. The Earth's diameter is about 12,700 kilometres. That gap is only about 0.16% of the diameter.
Because the Earth spins, it bulges a little around the equator. Even so, the difference in diameter is only about 0.34%. Once shrunk, it falls within what a ball's specification allows.
With big things, how bumpy they seem depends on what size you compare the bumps against. Let's shrink the Earth and check with real numbers.
How big is a mountain on a shrunken Earth?
A billiard ball is about 57 millimetres across. To shrink the Earth to that size, you divide every length by about 223 million. The shape stays the same. Everything just gets smaller by the same proportion.
Everest's height of 8,849 metres then becomes about 0.04 millimetres. A human hair is about 0.08 millimetres thick, so that is half a hair. Even the bottom of the deepest place, the Mariana Trench, becomes a dent of only about 0.05 millimetres.
Look at Figure 1. The right side is a greatly magnified view of the shrunken ball's surface. The round dot on the left is the cross-section of a hair, and next to it sits the triangle of Everest. You would hardly notice it by running a finger over it.
The equatorial bulge fits within the tolerance too
The other question is roundness. The Earth turns once a day, so it bulges outward around the equator. The diameter across the equator is about 43 kilometres longer than the diameter from the North Pole to the South Pole.
On the shrunken ball, that difference is about 0.19 millimetres. Billiard ball specifications are said to allow the diameter to be off by 0.127 millimetres either way, which is a range of about 0.25 millimetres. The Earth's distortion falls within that range.
Of course, a real ball's surface is polished far smoother than a 0.04-millimetre step. So "which is smoother?" depends on what you take as the measure of smoothness. What is certain is that the Earth is "a sphere so round you couldn't tell by running a finger over it."
On the shrunken ball, the average ocean depth is about 0.017 millimetres. Kitchen wrap is about 0.01 millimetres thick, so that is one or two sheets of it. The 99% of the air's weight sits in a layer about 0.13 millimetres thick. Clouds, rain and typhoons all happen inside this thin film.
The sea is thousands of kilometres wide but only a few kilometres deep. That is an even shallower proportion than water spread on a tray. So currents and winds mostly flow sideways. This is why a weather map can show them as sideways arrows on a flat map.
Summary
The gap between the Earth's highest and lowest points is only about 0.16% of its diameter. Shrink it to the size of a billiard ball and Everest is half the thickness of a hair, and the equatorial bulge fits within a ball's tolerance. The sea and air that protect us are a thin film, just a few sheets of kitchen wrap thick.
The Earth is so round you couldn't tell by running a finger over it.
We live inside a single sheet of wrap on its surface.
Why mountains can't grow any taller, and which mountain is farthest from the Earth's centre, are covered in "Is Everest Really the Tallest Mountain in the World?" For the weight of that thin film of air, see "Does Air Have Weight? How Many Kilos Is the Air in a Room?" And for a sense of distances in space, try "How Far Away Is the Moon?"
- Pick a ball at home and measure its diameter with a ruler (a ping-pong ball is about 40 millimetres, a football about 220 millimetres).
- Divide that diameter by the Earth's diameter of 12,742 kilometres. This is your shrinking ratio. Multiply it by Everest's 8.8 kilometres, or by 10 kilometres, the height airliners fly at.
- Compare the lengths you get with a hair (about 0.08 millimetres) or a sheet of paper (about 0.1 millimetres). Stick the paper on the ball's surface and think about how high that thickness would reach.
Even shrunk to a football, Everest is only about 0.15 millimetres, about the same as one sheet of paper. You can check the same sum with the slider in Figure 1.
For those who want more ― terms, formulas and links to textbooksEach part is labelled with its level, from middle-school science to university courses
- Middle schoolCovered in middle-school science and maths
- High schoolCovered in high-school "Earth science" and maths
- High school+Advanced high-school material, or textbook sidebar content
- UniversityUniversity specialist courses (geodesy and geophysics) not taught in high school
- ResearchTopics researchers are still studying, not yet taught even at university as settled
Middle schoolTerms: this phenomenon has names
- Similarity: changing only the size of a shape while keeping its form. When you shrink something, every length gets smaller by the same proportion.
- Scale: a number showing by what proportion real lengths were shrunk. It is the same idea as "1 to 25,000" on a map.
- Flattening: a number showing how squashed a sphere is. The Earth's is said to be about 1 in 298.
Middle schoolHigh schoolChecking with a formula: how many millimetres is a mountain when the Earth is shrunk to a billiard ball?
When you shrink something without changing its shape, every length gets smaller by the same proportion. This one property is enough to calculate the height of a mountain and the depth of the sea.
| In symbols | h′ ＝ h × ( d ÷ D ) |
| In words | Shrunken height ＝ real height × (shrunken ball's diameter ÷ Earth's diameter) |
| Where it comes from | The property of similarity. If you shrink a shape while keeping its form, the ratio of lengths is the same everywhere |
| Symbol | Meaning and unit |
|---|---|
| h | Real height or depth (millimetres) |
| h′ | Height or depth on the shrunken ball (millimetres) |
| d | Diameter of the shrunken ball (millimetres) |
| D | Diameter of the Earth (millimetres) |
| Earth's mean diameter | About 12,742 km (12742000000 mm) |
| Diameter of a billiard ball | About 57.2 mm |
| Height of Everest | 8849 m (8849000 mm) |
| Deepest point of the Mariana Trench | About 10,920 m (10920000 mm) |
| Average ocean depth | About 3700 m (3700000 mm) |
| Height holding 99% of the air's weight | Said to be about 30 km (30000000 mm) |
| Difference between equatorial and pole-to-pole diameters | About 43 km (43000000 mm) |
| Shrinking ratio (D ÷ d) | 12742000000 ÷ 57.2 ≒ 222760000 |
| Height of Everest (mm) | 8849000 ÷ 222760000 ≒ 0.040 |
| Depth of the Mariana Trench (mm) | 10920000 ÷ 222760000 ≒ 0.049 |
| Gap between highest and lowest points (mm) | 0.040 ＋ 0.049 ＝ 0.089 |
| That gap as a fraction of the diameter | 0.089 ÷ 57.2 ≒ 0.0016 |
| Equatorial bulge (diameter difference, mm) | 43000000 ÷ 222760000 ≒ 0.19 |
| Average ocean depth (mm) | 3700000 ÷ 222760000 ≒ 0.017 |
| Layer holding 99% of the air (mm) | 30000000 ÷ 222760000 ≒ 0.13 |
Everest is about 0.04 millimetres, half a hair (about 0.08 millimetres). The gap from mountain to trench is about 0.16% of the diameter, and the equatorial bulge of about 0.19 millimetres also fits within the ball tolerance range of about 0.25 millimetres. The sea is about 0.017 millimetres, as thin as one or two sheets of kitchen wrap.
High schoolHigh school+Why is the Earth round, and why is it slightly squashed?
High schoolOn a body as large as the Earth, gravity pulls rock toward the centre from every direction. A mountain that is too tall has foot rock that cannot bear the weight, and it sinks or collapses. So the larger a body is, the closer to a sphere it becomes. In high-school Earth science, you meet this in Eratosthenes' method, the first measurement of the Earth's size, and in the Earth ellipsoid.
High school+On a spinning Earth, the outward centrifugal force is greater near the equator. The Earth bulges toward the equator by that amount, and settles into a shape with a flattening of about 1 in 298. In the 17th and 18th centuries, there was a great debate over whether the Earth was fatter around the equator or longer from pole to pole. French survey expeditions settled it by measuring in Lapland and Peru.
UniversityThe Earth ellipsoid and the geoid
Geodesy describes the Earth's shape in two steps. The first is the Earth ellipsoid, a smooth ellipsoid of revolution used as an approximation (international standards include GRS80). The second is the geoid, a surface of equal gravitational potential, like sea level extended under the continents. The gap between the geoid and the Earth ellipsoid is said to be about 100 metres at most. On the shrunken ball, that is not even 0.0005 millimetres. Satellite measurements of the gravity field have revealed this fine shape in detail.
📖 Derivations of the formulas and further reading: Earth ellipsoid (Wikipedia, in Japanese) ／ Geoid (Wikipedia, in Japanese)
ResearchWhat we still don't know
- Most of the ocean floor is still a rough map. Only about a quarter of the seabed is said to have been measured in detail by ship sonar. The surfaces of the Moon and Mars are said to be mapped in finer detail.
- The Earth's flattening is changing little by little. When glaciers melt and water moves, or when land pushed down during an ice age rebounds, the flattening shifts slightly. Researchers continue to read the movement of ice and water from those changes.
- The cause of the geoid's bumps. South of the Indian Ocean, there is a spot where sea level is said to be about 100 metres lower than its surroundings. Uneven density deep inside the Earth is thought to be the cause, but the details are still debated.
So the content of this article is also "an explanation within what we know now." We are talking about a few microns, yet the seabed map it rests on is still unfinished.
Links to textbooks (by level)
| Level | Subject and unit | Where in this article |
|---|---|---|
| Middle school | Maths "Similar figures", social studies "Map scales" | The sum for shrinking the Earth to a billiard ball |
| High school | Basic Earth science "Shape and size of the Earth" | Why the Earth is round, Eratosthenes |
| High school+ | Earth science "Earth ellipsoid and flattening" | The equatorial bulge and 18th-century surveys |
| University | Geodesy and geophysics | The Earth ellipsoid and the geoid |
| Research | Seabed mapping, satellite gravity measurement | The seabed map, changes in flattening |
| ― | Connection to daily life | Feeling how thin a film the sea and air are |
- National Astronomical Observatory of Japan (国立天文台), ed., Chronological Scientific Tables (理科年表), Maruzen Publishing (Earth's size and shape, flattening)
- National Oceanic and Atmospheric Administration (NOAA), How deep is the ocean?
- Geospatial Information Authority of Japan (国土地理院), explanatory material on "The shape and size of the Earth" (geodetic reference system, GRS80)
- World Pool-Billiard Association, Equipment Specifications (ball diameter and tolerance)
- Report of The Nippon Foundation-GEBCO Seabed 2030 Project (share of the seabed that has been surveyed)
※This article is a general-audience science explainer. The figures given are rough guides to help you understand how things work. Billiard ball specifications may differ by organisation and era.