Calculation of the size of the Earth | Fantastic math and science wiki (2023)

The study of the size and shape of the Earth is known asgeodesyand has been practiced for centuries. Contrary to popular misconception, many mathematicians and historians knew that the Earthit was spherical. Technically, the shape of the Earth is called a "geoid", aEllipsoid shape. Possibly starting with Eratosthenes over 2,200 years ago, mathematicians tried to calculate the size of a spherical or elliptical Earth. The techniques of Eratosthenes and his predecessors are considered elementary, or at least simple, by today's mathematician, but they are impressive for their reach in calculating the entire Earth without being able to see or measure most of it, and for their accuracy.


  • Eratosthenes
  • Calculating the Size of the Earth by Eratosthenes
  • Additional historical estimates
  • references


Calculation of the size of the Earth | Fantastic math and science wiki (1) Portrait of Eratosthenes (276-195 BC). Mathematician, poet, philosopher, geographer.

One of the oldest and still most useful applications of mathematics is the use of geometric relationships to derive the dimensions of things without measuring them directly. About 2,000 years ago, an ancient Greek scholar, philosopher, poet and mathematician named Eratosthenes used incredibly simple geometry to calculate the size of the Earth with very high precision. The ancient Greeks were known for their ability to use geometry to describe the magnitude of things they couldn't directly perceive. His geometry was always correct and intelligent, but his conclusions were often off target. (Due to inaccuracies in the measurements used in the calculations.) Earth's scale is a measurement they may have come very close to.

Eratosthenes of Alexandria (276-195 BC) was born in Cyrene (present-day Libya), studied in Athens, and became chief librarian of the Library of Alexandria. The Library of Alexandria was the center of knowledge in the Hellenistic world. Scholars of antiquity were not bound by specializations. Eratosthenes first became famous as a poet, invented the first system of degrees of latitude and longitude, calculated the tilt of the Earth's axis, invented the discipline of geography, developed aAlgorithm to calculate prime numbers, and was the first person to attempt to create an accurate map of the world based on physical data. Supposedly, his classmates gave him the nicknameB\BetaB, because he had a reputation for being at least second best in the world at just about everything.

Calculation of the size of the Earth | Fantastic math and science wiki (2) World map of Eratosthenes

The above map was compiled from land surveys and sky observations made during the time of Alexander the Great's conquest. It covers an extremely large portion of the earth, regardless of when it was created. Much of the information on the map was probably transmitted by word of mouth and by compiling many observations made by many people over a long period of time. It also does a relatively impressive job of showing the approximate shapes and directions of landmasses, oceans, and rivers.

From his map, it can be concluded that Eratosthenes had no first-, second-, or third-hand knowledge of an entire hemisphere of the earth and most of Asia and Africa. However, using simple geometry, he was able to deduce the size of the Earth, and therefore the size of the planet, which he did not know.

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Calculating the Size of the Earth by Eratosthenes

He observed that at noon on the summer solstice in the northern hemisphere, the sun's rays fell on the Earth at different angles in two different places. In the city of ancient Syene (present-day Aswan, Egypt), at noon on the solstice, the sun was directly overhead and cast no shadows on objects below (Aswan is very close to the sun).Tropic of Cancer. This was demonstrated by observing the sun fully illuminating the bottom of a very deep shaft at Syene at noon. The angle between the sun's rays and objects perpendicular to the Earth's surface was practically zero.

In Alexandria, the sun cast light shadows behind objects around noon the same day. Measuring the dimensions of shadows and the objects that cast them. Eratosthenes was able to calculate that the sun's rays are at an angle of Alexandria150\frac1{50}501of a circle (about 0.13 radians or 7.2 degrees). He knew the distance was between Alexandria and Syene500050005000“Stadiums” of topographic surveys carried out between the two cities. A furlong was an ancient Greek unit of measurement (150-200 meters). He also assumed that Syene was south of Alexandria and on the same longitude. This would mean that the distance between the two cities represents an arc of the Earth's circumference (in fact, Aswan is about three degrees of longitude east of Alexandria).

Calculation of the size of the Earth | Fantastic math and science wiki (3) A cross-section of Earth at noon on the summer solstice in the northern hemisphere.

Eratosthenes assumed that the sun was so far from the earth that its rays would strike the earth parallel to each other. in the diagram aboveAStore AAis the angle at which the sun's rays strike the earth in Alexandria at noon on the solstice.Bstore BBrepresents the angle subtended at the center of the earth by the arc representing the distance between Alexandria and Syene. From the equivalence of alternate interior angles through a transversal line to two parallel lines,A=B=1/50\Store A = \Store B = 1/50A=B=1/50of a circle Since the angle is B1/501/501/50of a circle and draws an arc on the earth's surface of 5000 stadia, which would be the circumference of the earth5,000×50=250,0005000 x 50 = 250 0005,000×50=250,000stages.

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Since there is academic debate over what the definition of "stadium" is, there is no way of knowing exactly how close Eratosthenes was to the currently accepted value for the Earth's circumference. A "stadium" is believed to be defined as the length of an ancient Greek stadium used for athletic contests. Most of them are about110\frac1{10}101than a mile, but some are longer and some are shorter. There is evidence that the word Stade had different meanings in different contexts, at different times and places. It is not certain which definition of stadium Eratosthenes used. The actual circumference of the Earth is40,07540.07540,075kilometers at the equator.

Calculation of the size of the Earth | Fantastic math and science wiki (4) Variation of Eratosthenes' estimates by stadium length

Separately, he calculated the size of Earth to within an order of magnitude and possibly very close. Perhaps you were also lucky. The errors may be due to: a) the fact that Syene and Alexandria are not exactly on the same longitude; b) significant errors in measuring the land distance between Alexandria and Syene; or c) measuring the angles cast by shadows in Alexandria.

It is a common modern misconception that past civilizations believed in a flat Earth of unknown dimensions. Eratosthenes' calculation spread far and wide and was preserved through successive generations and geographically diverse civilizations. This estimation of the size of the Earth and its method would remain canonical for the educated elite of future civilizations in Europe, North Africa, the Middle East and South Asia well into the 17th century.AND^\texto{th}ANDCentury, when more precise measurements could be made to limit the size of the Earth.

Additional historical estimates

Arya Bhatta
Aryabhata was an Indian mathematician who lived between 476 and 550 AD and accurately wrote that the Earth is spherical. He also estimated the Earth's circumference at 67 miles from modern measurements, with an error of only about 0.27%.[1]

Abu Arrayhan Muhammad ibn Ahmad al-Biruni
Al-Biruni, a Persian mathematician who lived between 973 and 1048 AD, used triangulation to measure the land. His estimate was that the Earth had a radius of 6339.6 km, 16.8 km less, about 0.27% less than modern measurements.[2]

Robert Helmert
In 1906, German geodesist Friedrich Robert Helmert determined that the Earth is a global ellipsoid and calculated it to be accurate to within 100 meters, 0.002% of modern measurements.

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  1. O'Connor, J. e Robertston, E..Aryabhata el-Viejo. Retrieved March 23, 2016 from
  2. O'Connor, J. e Robertston, E..Abu Arrayhan Muhammad ibn Ahmad al-Biruni. Retrieved March 23, 2016 from


How did we calculate the size of the Earth? ›

The Greek philosopher Aristotle (384-322 B.C.) is credited as the first person to try and calculate the size of the Earth by determining its circumference (the length around the equator) He estimated this distance to be 400,000 stades (a stadia is a Greek measurement equaling about 600 feet).

Who calculated the size and shape of the earth? ›

The first person to determine the size of Earth was Eratosthenes of Cyrene, who produced a surprisingly good measurement using a simple scheme that combined geometrical calculations with physical observations. Eratosthenes was born around 276 B.C., which is now Shahhat, Libya.

What is the science of measuring the size and shape of the earth? ›

Geodesy is the science of measuring and monitoring the size and shape of the Earth, including its gravity field, and determining the location of points on the Earth's surface.

How do we calculate the circumference of the Earth? ›

The circumference of the Earth can be calculated using the formula, Circumference = 2 πR or π D, where R is the radius of Earth and D is the diameter. By knowing the Equatorial diameter or the polar diameter of the Earth, the circumference can be found.

What is the size of Earth answer? ›

Its diameter (the distance from one side to the other through Earth's center) is 7,926 miles (about 12,756 kilometers). Earth is slightly smaller when measured between the North and South Poles which gives a diameter of 7,907 miles (12,725 kilometers).

What is the exact size of Earth? ›

The diameter of a circle is twice the radius, giving us a diameter for Earth of 12,756 km. Note: The Earth is almost, but not quite, a perfect sphere. Its equatorial radius is 6378 km, but its polar radius is 6357 km - in other words, the Earth is slightly flattened.

Who named Earth? ›

All of the planets, except for Earth, were named after Greek and Roman gods and godesses. The name Earth is an English/German name which simply means the ground. It comes from the Old English words 'eor(th)e' and 'ertha'.

Who named our planet Earth? ›

Unlike the other planets in the Solar System, in English, Earth does not directly share a name with an ancient Roman deity. The name Earth derives from the eighth century Anglo-Saxon word erda, which means ground or soil, and ultimately descends from Proto-Indo European *erþō.

Who proved that he can figure out the size of the Earth through the information gathered during lunar eclipse? ›

Measuring the Size of the Earth

In 3rd Century B.C., Aristarchus of Samos reasoned he could figure out the size of the Earth based on information available during a lunar eclipse.

What is the shape of the Earth question and answer? ›

Earth is an oblate spheroid. This means it is spherical in shape, but not perfectly round. It is slightly bulged at the equator and is flattened at the poles.

What is the math of Earth? ›

The Mathematics of Planet Earth (MPE) research group develops mathematical modelling tools for studying Planet Earth, its life-supporting capacity, and the impact of human activities.

What are 3 common measurements found in science? ›

The three most common base units in the metric system are the meter, gram, and liter. The meter is a unit of length equal to 3.28 feet; the gram is a unit of mass equal to approximately 0.0022 pounds (about the mass of a paper clip); and the liter is a unit of volume equal to 1.05 quarts.

How did Aristotle calculate the circumference of the Earth? ›

By measuring the length of the shadow in Alexandria at noon on the Summer Solstice when there was no shadow in Syene, he could measure the circumference of the Earth! At Syene: The Sun is directly overhead, no shadows are cast at that moment.

What is the formula for area and circumference? ›

Use the formulas to calculate the circumference and area: c = 2πr and A = πr² . The circumference should equal c = 2π × 8 cm = 50.265 cm . The area should equal A = π × (8 cm)² = 201.06 cm² . To check your results, input the 8 cm diameter in the calculator.

What is Earth very short answer? ›

Earth, our home planet, is a world unlike any other. The third planet from the sun, Earth is the only place in the known universe confirmed to host life. With a radius of 3,959 miles, Earth is the fifth largest planet in our solar system, and it's the only one known for sure to have liquid water on its surface.

What is Earth small answer? ›

The Earth is a rocky planet that revolves around the sun in the third orbit, after Mercury and Venus. It is located at a distance from the Sun which provides it with the 'just right' temperature to harbour liquid water on its surface. It also has an atmosphere that contains Nitrogen and Oxygen in the majority.

How did scientists calculate the radius of Earth? ›

During the 3 century, B.C. Eratosthenes was able to mathematically calculate the diameter of the earth by comparing the difference in the angle of the sun's rays at different geographic points. He was the first person that was able to calculate the radius of the earth.

What is the real name of Earth? ›

It is a common misconception that “Terra” is the internationally-recognized scientific name of the planet, but in reality Earth does not have an official international name. The standard English name of the planet, including in science, is “Earth”.

Is Earth named after any God? ›

Earth is the only planet in our solar system not named after a Greco-Roman deity. The name used in Western academia during the Renaissance was Tellus Mater or Terra Mater, the Latin for “earth mother”, i.e. “Mother Earth”, goddess of the earth in ancient Roman religion and mythology.

Which planet is the God of heaven? ›

Uranus, in Greek mythology, the personification of heaven.

What was the Earth called before? ›

This giant landmass known as a supercontinent was called Pangea. The word Pangaea means "All Lands", this describes the way all the continents were joined up together. Pangea existed 240 million years ago and about 200 millions years ago it began to break apart.

Who gave the shape of Earth? ›

Although there was some evidence to support a spherical Earth, it wasn't until the 3rd century BC that a Greek mathematician named Eratosthenes calculated the circumference of the planet with remarkable accuracy. He is credited as “the Greek who proved the earth was round”.

When was the shape of the Earth discovered? ›

The earliest documented mention of the concept dates from around the 5th century BC, when it appears in the writings of Greek philosophers. In the 3rd century BC, Hellenistic astronomy established the roughly spherical shape of Earth as a physical fact and calculated the Earth's circumference.

Who was the first person on Earth? ›

Adam is the name given in Genesis 1-5 to the first human. Beyond its use as the name of the first man, adam is also used in the Bible as a pronoun, individually as "a human" and in a collective sense as "mankind".

How old is the Earth? ›

Earth is estimated to be 4.54 billion years old, plus or minus about 50 million years. Scientists have scoured the Earth searching for the oldest rocks to radiometrically date. In northwestern Canada, they discovered rocks about 4.03 billion years old.

What shape is the world made of? ›

In a new paper in the Proceedings of the National Academy of Sciences, a team from the University of Pennsylvania, Budapest University of Technology and Economics, and University of Debrecen uses math, geology, and physics to demonstrate that the average shape of rocks on Earth is a cube.

What did Earth look like when it was first made? ›

In Earth's Beginning

At its beginning, Earth was unrecognizable from its modern form. At first, it was extremely hot, to the point that the planet likely consisted almost entirely of molten magma. Over the course of a few hundred million years, the planet began to cool and oceans of liquid water formed.

How did scientists figure out how old Earth is? ›

They have examined rocks from the moon and from meteorites, neither of which have been altered by the rock cycle. The same techniques of radiometric dating have been used on those rocks. All the data from Earth and beyond has led to the estimated age of 4.5 billion years for our planet.

What best explains the shape of the earth? ›

The oblate spheroid, or oblate ellipsoid, is an ellipsoid of revolution obtained by rotating an ellipse about its shorter axis. It is the regular geometric shape that most nearly approximates the shape of the Earth. A spheroid describing the figure of the Earth or other celestial body is called a reference ellipsoid.


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