Perimeter and circumference

Summary: Measuring perimeter and circumference is a geometric task with a long history of methods.

Measurements of length and distance abound in daily life, from the height of a child to the distance from home to the store. Perimeter and circumference are types of length measurements. The perimeter of a geometric entity is the path that surrounds its area. The word derives from its Greek roots peri (meaning “around”) and from meter (meaning “measure”). In stricter mathematical sense, perimeter is defined as the length of the curve constituting the boundary of a two-dimensional, planar closed surface.

For example, the perimeter of a square whose side measures length a is 4a. Perimeter is important for applications such as landscaping projects, construction, and building fences. Circumference is defined as the perimeter of a circle. The circumference of a circle of radius (r) is 2πr. The circumference of a circle has played a very important role throughout history in the approximation of the mathematical constant π, which was defined as the ratio of the circumference (C) of the circle to its diameter (d). Perhaps the most common reference to circumference that most people encounter regularly is the circumference of one’s waist—the size of their waist.

The waist circumference is used as a measurement for some clothing and is also associated with type II diabetes, dyslipidemia, hypertension, and other cardiovascular diseases. Students investigate perimeter beginning in primary school, and middle grade students explore circumference. Students formulate the length of general curves, referred to as the “arc length” or “rectification,” as integrals in calculus courses.

History

There is a long history of computations involving the perimeter or circumference of figures. One way was to measure length was with ropes. For instance, statements about rope measurements and the Pythagorean theorem can be found in Katyayana’s Sulbasutra. Another way was to compare the length of two figures. A Babylonian clay tablet was discovered in 1936 and was noted as relating the hexagon perimeter to 0;57,36 (in base 60), or 24/25 times the circumference of a circumscribed circle. Mathematicians like Archimedes of Syracuse estimated the circumference or a value for π by using the perimeters of inscribed and circumscribed polygons with many sides. For example, Archimedes was known to have used 96-sided polygons. Mahavira estimated the circumference of an ellipse. In ancient times, the semiperimeter, or half the perimeter, was useful in computing many geometrical properties of polygons such as altitude, exradius, and inradius of a triangle. The semiperimeter also appears in Heron of Alexandria’s formula for the area of a triangle. The semiperimeter of a rectangle is the sum of the length plus the height and is noted as appearing on Babylonian clay tablets. Brahmagupta used the semiperimeter of a quadrilaterial in the computation of its area.

The circle is a special geometric figure, for it is the curve, given a fixed perimeter, which encompasses the maximum surface area. This is known as the isoperimetric problem. Proclus commented that, “a misconception is held by geographers who infer the size of a city from the length of its walls.” The Babylonians may have worked on related problems in their investigations of solutions to quadratic equations generated by the setting of the semiperimeter and area to constants. The isoperimetric problem was partially solved by the Greek mathematician Zenodorus.

Pappus of Alexandria compared the areas of figures with a fixed perimeter. In the tenth century, Abu Jafar al-Khazin proved that an equilateral triangle has greater area than isosceles or scalene triangles of the same fixed perimeter. Many mathematicians worked on the isoperimetric problem using a variety of techniques including methods from geometry, analysis, vectors, and calculus. In 1842, a German mathematician named Jakob Steiner used geometric arguments to present five proofs of the theorem. However, Steiner had assumed that a solution was possible, which was a subtle flaw to otherwise creative arguments. Karl Weierstrass proved the existence of such solutions in 1879. Other mathematicians proved the results in a variety of other ways.

Historical Applications and Computations

One application of circumference of a circle is the computation of the Earth’s circumference. Eratosthenes of Cyrene, in 240 b.c.e., computed the Earth’s circumference using trigonometry and the angle of elevation of the sun at noon in Alexandria and Syene. He made an assumption that the Earth and the sun were perfect spheres and that the sun was so far away that its rays hitting the Earth could be considered parallel. By measuring the shadows thrown by sticks on the summer solstice, Eratosthenes derived a formula to measure the circumference of the Earth and determined it to be 252,000 stadia. Teachers in mathematics classrooms share Eratosthenes’s calculation in order to highlight his ingenuity and showcase the power of setting up proportions and applying the congruence of alternate interior angles of parallel lines. There is debate about the value of a stadia, but historians estimate that Eratosthenes was correct within a 2% to 15% margin of error. The Indian mathematician Aryabhata made revolutionary contributions toward the understanding of astronomy at the turn of the fifth century. His calculations on π, the circumference of Earth, and the length of the solar day were remarkably close approximations.

The middle of the seventeenth century marked a fruitful time in the history of calculating the length of general curves. For instance, the curve that forms the shape of a nautilus shell is called the “logarithmic spiral” or “equiangular spiral.” Evangelista Torricelli described its length using geometric methods. Christopher Wren published the rectification of the cycloid curve. Hendrik van Heuraet and Pierre de Fermat independently explored ideas that would eventually lead to the integral formula of arc length.

In the twentieth century, methods from fractals, popularized by Benoit Mandelbrot, have proven useful in modeling objects like a coastline. One example that is regularly examined in mathematics classrooms is the Koch snowflake, named for Helge von Koch, an example of a curve that bounds a region with finite area yet has infinite perimeter.

Bibliography

Blasjo, Viktor. “The Isoperimetric Problem.” The American Mathematical Monthly 112, no. 6 (2005).

Briggs, William. “Lessons From the Greeks and Computers.” Mathematics Magazine 55, no. 1 (1982).

Dunham, William. “Heron’s Formula for Triangular Area.” In Journey through Genius: The Great Theorems of Mathematics. Hoboken, NJ: Wiley, 1990.

Steinhaus, H. Mathematical Snapshots. 3rd ed. New York: Dover, 1999.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991.