Measurements of area

Summary: Measuring area is an important mathematical calculation that has been studied for thousands of years.

“Area” is often thought of as the amount of a plane that a two-dimensional figure occupies. The name comes from the Latin word area, which means a vacant piece of level ground, reflecting the fact that formulas to calculate units of area, were often developed to facilitate surveying and the calculation of the size of land plots for tax or other purposes. Some formulas to perform area calculations for simple geometric shapes were known in ancient times, while other calculations like the area of curved figures could only be approximated before the development of calculus and a greater depth of understanding regarding the constant π.

In the twenty-first century, primary school students explore attributes such as area and how it changes when the shape of an object changes. They also use formulas to find the areas of rectangles, triangles, and parallelograms and investigate the surface areas of rectangular solids. In the middle grades, the surface areas of prisms, pyramids, and cylinders are also a focus. In high school, students calculate the area and surface area of various geometric figures such as cones, spheres, and cylinders. In calculus classes, students develop a deeper understanding of area through integration techniques.

Ancient History: Egypt, Babylon, and India

The Moscow Mathematical Papyrus (c. 1850 b.c.e.) and the Ahmes, or Rhind, Papyrus (c. 1650 b.c.e.) provide evidence that Egyptians of this period had systems for calculating the areas of different geometric shapes, including triangles, rectangles, and circles. The approach to these calculations is frequently expressed using methods based on the interrelationship between different geometric figures. For instance, one problem in the Ahmes Papyrus notes that the area of a circular field of diameter 9 is the same as the area of a square field with a side of length 8. Historians of mathematics have converted these calculations to an estimate of π that is about 3 1/6 (compared to the correct value of 3.141592…). No differentiation is made between exact and approximate formulas, and there is nothing resembling a proof or theorem in the modern sense—the papyrus presents ways to perform calculations.

The ancient Babylonians also had methods, preserved on clay tablets written in cuneiform, for calculating area. Historians have these tablets and inferred from the calculations values of π, such as 3 and 3.125. As with the Egyptians, methods to calculate area were often expressed by the relation between different geometric figures, and there is no evidence of proofs.

Practical needs also motivated the mathematics presented in the Sulbasutras, appendices to the Vedas (Hindu scriptures), which explain how to construct sacrificial altars. These scriptures include methods for constructing circles from squares and vice versa, indicating 577/408 as an approximation of √2.

Ancient Greeks

The ancient Greeks were able to estimate or derive many areas, in some cases building upon earlier work done by people from other cultures and civilizations. Antiphon the Sophist (480–411 b.c.e.), Eudoxus of Cnidus (408–355 b.c.e.), Archimedes of Syracuse (287–212 b.c.e.), and others approximated the area of figures like the circle by using inscribed and circumscribed polygons, a technique referred to as the “method of exhaustion.” Using polygons, Archimedes was also able to show that the surface area of a sphere is four times the area of “the greatest circle in it.” In mathematics classrooms, students may discover this relationship by peeling an orange and fitting the peel pieces into four circles that have the same diameter as the equator of the orange. Expressed in modern terminology, each circle has area πr2 where r is the radius, so the surface area is 4πr2. In ancient Greece, the Pythagorean theorem, named for Pythagoras of Samos, was expressed in terms of areas of squares rather than the lengths of the sides of a right triangle; the square figure on the hypotenuse had the same area as the sum of the other two squares. Euclid of Alexandria (325–265 b.c.e.) collected the theorems of Pythagoras and other predecessors into the treatise now known as the Elements, which has proven to be one of the most influential mathematical textbooks in history. Heron of Alexandria (10–70 c.e.) published the Metrica, a treatise that collected formulas for calculating the area and volume of many different geometric figures and also presented what is known today as “Heron’s formula” for expressing the area of a triangle in terms of its sides:

where K is the area of the triangle, a, b, and c are the length of the sides, and s is the semiperimeter (half the sum of the lengths of the sides).

Seventeenth Century

In the seventeenth century, German astrologer and mathematician Johannes Kepler applied the ideas of calculus to compute the area and volume of conic sections and casks. Reportedly, his interest was sparked while at his own wedding reception; Kepler became interested in finding a method for calculating the volume of wine in barrels that were not perfect cylinders (they were wider in the middle than at the top and bottom), meaning that the simple formula for the volume of a cylinder could not be applied. Development of differential and integral calculus, necessary to find the area of curved figures, is attributed to both German mathematician Gottfried Leibniz and English mathematician Sir Isaac Newton, in the seventeenth century. Also in the seventeenth century, the French mathematician Albert Girard published a treatise that was the first to use the abbreviations “sin,” “cos,” and “tan” for the sine, cosine, and tangent and demonstrated that the area of a spherical triangle depends on its interior angles, which is known as “Girard’s Theorem.”

Recent Developments

In the nineteenth century, the ancient area problem of squaring the circle was finally resolved. The challenge had been to construct a square that had the same area as a circle using only a ruler and compass. In 1882, German mathematician Ferdinand von Lindemann proved that π is a transcendental number, meaning that it is not equal to any finite sequence of algebraic operations on integers. This characteristic also meant that ruler-and-compass methods of constructing a square with area the same as the area of a circle of radius 1 were also doomed to failure. However, methods other than ruler and compass constructions can be used to obtain such a square.

In 1917, the Japanese mathematician Soichi Kakeya posed what is known as the “Kakeya problem,” which asks whether there is a minimum region in a plane in which a needle (line segment) can be freely rotated. This area minimization problem was solved in 1927 by Russian mathematician Abram Samoilovitch Besicovitch and also in 1928 by German mathematician Oskar Perron. In the 1930s, American mathematician Jesse Douglas and Hungarian Tibor Rado published solutions to “Plateau’s Problem,” which requires finding the area of a minimal surface bounded by a curve. The problem is named for nineteenth-century Belgian physicist Joseph Plateau, although it was first posed in the eighteenth century by Joseph-Louis Lagrange.

Bibliography

Boyer, Carl. B. A History of Mathematics. 2nd ed. Rev. Uta C. Merzbach. Hoboken, NJ: Wiley, 1991.

Darling, David. The Universal Book of Mathematics: From Abracadabra to Zeno’s Paradoxes. Hoboken, NJ: Wiley, 2004.

National Council of Teachers of Mathematics. “Principles & Standards for School Mathematics: Higher Standards for Our Students… Higher Standards for Ourselves.” http://standards.nctm.org.

Washington State Department of Transportation. “The Metrics International System of Units.” http://www.wsdot.wa.gov/reference/metrics/factors.htm.

Zebrowski, Ernest, Jr. A History of the Circle: Mathematical and the Physical Universe. New Brunswick, NJ: Rutgers University Press, 1999.