Finding the Circumference of the Earth
by aoum, Mar 8, 2025, 12:32 AM
Finding the Circumference of the Earth: A Mathematical Approach
How big is the Earth? Long before satellites and GPS, ancient mathematicians devised ingenious methods to estimate its circumference. The most famous method was developed over 2,000 years ago by the Greek mathematician Eratosthenes, using nothing but shadows, angles, and basic geometry. In this post, we’ll explore how he did it, derive the formulas, and apply modern techniques to verify his result.
1. Eratosthenes’ Method (Around 240 BCE)
Eratosthenes, the chief librarian of the Library of Alexandria, made a remarkable observation:
At noon on the summer solstice in Syene (modern-day Aswan, Egypt), the Sun was directly overhead, meaning a vertical stick would cast no shadow. However, in Alexandria, about 800 km north, a vertical stick did cast a shadow.
He realized that this happened because the Earth is curved. By measuring the shadow’s angle and knowing the distance between the two cities, he estimated the Earth’s circumference.
Step 1: Measure the Shadow Angle
Eratosthenes measured that in Alexandria, the angle of the Sun’s rays relative to the vertical was about
.
This angle corresponds to the central angle of an arc between Alexandria and Syene.
Step 2: Use Proportions of a Circle
Since a full circle is
, the fraction of the Earth's circumference covered by the distance from Alexandria to Syene is:
![\[
\frac{\theta}{360^\circ} = \frac{d}{C}
\]](//latex.artofproblemsolving.com/6/1/9/619eba09eb538986950748f219bdcbb2ceab58aa.png)
where:
Rearranging for
:
![\[
C = \frac{d \times 360^\circ}{\theta}
\]](//latex.artofproblemsolving.com/4/f/a/4fa8fc8d8d5db54680cf5b85134d42bc7cf330ae.png)
Substituting the values:
![\[
C = \frac{800 \times 360}{7.2} \approx 40,000 \text{ km}.
\]](//latex.artofproblemsolving.com/2/9/6/2968fc38090ba19d9d8e7c6542b166f279001caa.png)
This is remarkably close to the modern accepted value of about 40,075 km at the equator!
2. Mathematical Derivation Using Arc Length
A more general way to derive the circumference using any two points on the Earth's surface is through the arc length formula:
![\[
s = r\theta
\]](//latex.artofproblemsolving.com/1/f/6/1f6d5de442b648c502a8ef82c6531c0cfbcbccca.png)
where:
Solving for
:
![\[
r = \frac{s}{\theta}
\]](//latex.artofproblemsolving.com/e/a/c/eac7ad2cf4b2adb48ad86252c4b0429950a9bd00.png)
The full circumference is given by:
![\[
C = 2\pi r.
\]](//latex.artofproblemsolving.com/0/8/a/08a5700b80b197abf7bb4a33d8a427682afbc883.png)
Substituting
:
![\[
C = 2\pi \times \frac{s}{\theta}.
\]](//latex.artofproblemsolving.com/4/2/3/423882a2da101b8df05656743ee6c9da4cc494c7.png)
Using Eratosthenes' values:
![\[
r = \frac{800}{\frac{7.2\pi}{180}} = \frac{800 \times 180}{7.2\pi} \approx 6,366 \text{ km}.
\]](//latex.artofproblemsolving.com/4/c/a/4ca39b001f93727e4591edf7ed501fffb9ec3479.png)
Then, the circumference is:
![\[
C = 2\pi (6,366) \approx 40,000 \text{ km}.
\]](//latex.artofproblemsolving.com/7/8/6/786af83138a0a3f0fbc0cbb95e977af2ec73eaa1.png)
3. Modern Verification Using Satellite Data
Today, we can verify the Earth's circumference using satellite measurements:
A modern way to compute Earth's circumference using geodetic data involves measuring the equatorial radius
km and applying:
![\[
C = 2\pi r_e = 2\pi (6,378.1) \approx 40,075 \text{ km}.
\]](//latex.artofproblemsolving.com/9/a/c/9accb12c388a98a0ad222514eacbedcb008ca258.png)
4. Other Methods to Measure Earth's Circumference
a) Using GPS Coordinates
Given two locations with latitude difference
, the circumference can be approximated using:
![\[
C = \frac{d}{\Delta \phi} \times 360^\circ
\]](//latex.artofproblemsolving.com/c/7/9/c791f845c15491a7928a5495161fa0580ba12d73.png)
where
is the north-south distance and is in degrees.
b) Using Pendulum Measurements
Gravity varies slightly with latitude due to the Earth's shape. By measuring the acceleration due to gravity at different latitudes, one can estimate the Earth's radius and circumference.
c) Using Airplane Speeds
An aircraft traveling at a known speed along a great circle route can estimate the Earth's circumference by measuring the total distance covered.
5. Why Does This Matter?
The measurement of Earth’s circumference has profound implications:
6. Conclusion: Ancient Genius Meets Modern Science
Eratosthenes’ experiment remains one of the greatest achievements of ancient science. His method, based on simple observations and proportional reasoning, produced an answer remarkably close to modern measurements. Today, with advanced technology, we continue to refine our understanding of Earth’s shape and size, but his fundamental insight still holds true.
References
How big is the Earth? Long before satellites and GPS, ancient mathematicians devised ingenious methods to estimate its circumference. The most famous method was developed over 2,000 years ago by the Greek mathematician Eratosthenes, using nothing but shadows, angles, and basic geometry. In this post, we’ll explore how he did it, derive the formulas, and apply modern techniques to verify his result.
1. Eratosthenes’ Method (Around 240 BCE)
Eratosthenes, the chief librarian of the Library of Alexandria, made a remarkable observation:
At noon on the summer solstice in Syene (modern-day Aswan, Egypt), the Sun was directly overhead, meaning a vertical stick would cast no shadow. However, in Alexandria, about 800 km north, a vertical stick did cast a shadow.
He realized that this happened because the Earth is curved. By measuring the shadow’s angle and knowing the distance between the two cities, he estimated the Earth’s circumference.
Step 1: Measure the Shadow Angle
Eratosthenes measured that in Alexandria, the angle of the Sun’s rays relative to the vertical was about
.This angle corresponds to the central angle of an arc between Alexandria and Syene.
Step 2: Use Proportions of a Circle
Since a full circle is
, the fraction of the Earth's circumference covered by the distance from Alexandria to Syene is:![\[
\frac{\theta}{360^\circ} = \frac{d}{C}
\]](http://latex.artofproblemsolving.com/6/1/9/619eba09eb538986950748f219bdcbb2ceab58aa.png)
where:
- = distance between Alexandria and Syene
km
, - = circumference of the Earth,
- .
k
,Rearranging for
:![\[
C = \frac{d \times 360^\circ}{\theta}
\]](http://latex.artofproblemsolving.com/4/f/a/4fa8fc8d8d5db54680cf5b85134d42bc7cf330ae.png)
Substituting the values:
![\[
C = \frac{800 \times 360}{7.2} \approx 40,000 \text{ km}.
\]](http://latex.artofproblemsolving.com/2/9/6/2968fc38090ba19d9d8e7c6542b166f279001caa.png)
This is remarkably close to the modern accepted value of about 40,075 km at the equator!
2. Mathematical Derivation Using Arc Length
A more general way to derive the circumference using any two points on the Earth's surface is through the arc length formula:
![\[
s = r\theta
\]](http://latex.artofproblemsolving.com/1/f/6/1f6d5de442b648c502a8ef82c6531c0cfbcbccca.png)
where:
= arc length (distance between two locations along the surface),- = Earth's radius,
= central angle in radians.
= arc length (distance between two locations along the surface),
= central angle in radians.Solving for
:![\[
r = \frac{s}{\theta}
\]](http://latex.artofproblemsolving.com/e/a/c/eac7ad2cf4b2adb48ad86252c4b0429950a9bd00.png)
The full circumference is given by:
![\[
C = 2\pi r.
\]](http://latex.artofproblemsolving.com/0/8/a/08a5700b80b197abf7bb4a33d8a427682afbc883.png)
Substituting
:![\[
C = 2\pi \times \frac{s}{\theta}.
\]](http://latex.artofproblemsolving.com/4/2/3/423882a2da101b8df05656743ee6c9da4cc494c7.png)
Using Eratosthenes' values:
km,
radians,
km,
radians,![\[
r = \frac{800}{\frac{7.2\pi}{180}} = \frac{800 \times 180}{7.2\pi} \approx 6,366 \text{ km}.
\]](http://latex.artofproblemsolving.com/4/c/a/4ca39b001f93727e4591edf7ed501fffb9ec3479.png)
Then, the circumference is:
![\[
C = 2\pi (6,366) \approx 40,000 \text{ km}.
\]](http://latex.artofproblemsolving.com/7/8/6/786af83138a0a3f0fbc0cbb95e977af2ec73eaa1.png)
3. Modern Verification Using Satellite Data
Today, we can verify the Earth's circumference using satellite measurements:
- The equatorial circumference is about 40,075 km.
- The polar circumference is slightly less, about 40,008 km, due to the Earth's slight flattening at the poles.
A modern way to compute Earth's circumference using geodetic data involves measuring the equatorial radius
km and applying:![\[
C = 2\pi r_e = 2\pi (6,378.1) \approx 40,075 \text{ km}.
\]](http://latex.artofproblemsolving.com/9/a/c/9accb12c388a98a0ad222514eacbedcb008ca258.png)
4. Other Methods to Measure Earth's Circumference
a) Using GPS Coordinates
Given two locations with latitude difference
, the circumference can be approximated using:![\[
C = \frac{d}{\Delta \phi} \times 360^\circ
\]](http://latex.artofproblemsolving.com/c/7/9/c791f845c15491a7928a5495161fa0580ba12d73.png)
where
is the north-south distance and is in degrees.b) Using Pendulum Measurements
Gravity varies slightly with latitude due to the Earth's shape. By measuring the acceleration due to gravity at different latitudes, one can estimate the Earth's radius and circumference.
c) Using Airplane Speeds
An aircraft traveling at a known speed along a great circle route can estimate the Earth's circumference by measuring the total distance covered.
5. Why Does This Matter?
The measurement of Earth’s circumference has profound implications:
- Navigation: Helps in map-making and GPS technology.
- Space Exploration: Essential for satellite orbits and planetary studies.
- Scientific History: Demonstrates how mathematical reasoning can uncover fundamental truths.
- Flat Earth Debunking: Provides solid empirical evidence for Earth's curvature.
6. Conclusion: Ancient Genius Meets Modern Science
Eratosthenes’ experiment remains one of the greatest achievements of ancient science. His method, based on simple observations and proportional reasoning, produced an answer remarkably close to modern measurements. Today, with advanced technology, we continue to refine our understanding of Earth’s shape and size, but his fundamental insight still holds true.
References
- Wikipedia: Eratosthenes
- Maor, E. The Pythagorean Theorem: A 4,000-Year History (2007).
- Lang, S. A First Course in Calculus (2005).
- Kline, M. Mathematical Thought from Ancient to Modern Times (1972).
March 2025