circle

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Understand how to calculate a circle's area with the help of chain and ruler

Eccentricity of conic sectionsThe eccentricity of a conic section completely characterizes its shape. For example, all circles have zero eccentricity, and all parabolas have unit eccentricity; hence, all circles (and all parabolas) have the same shape, only varying in size. (Under appropriate magnification they are indistinguishable.) In contrast, ellipses and hyperbolas vary greatly in shape.

The transformation of a circular region into an approximately rectangular regionThis suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

Figure 11: Beltrami's circle model.

concentric circles and infinity

Projective conic sectionsThe conic sections (ellipse, parabola, and hyperbola) can be generated by projecting the circle formed by the intersection of a cone with a plane (the reality plane, or RP) perpendicular to the cone's central axis. The image of the circle is projected onto a plane (the projective plane, or PP) that is oriented at the same angle as the cutting plane (Ω) passing through the apex (“eye”) of the double cone. In this example, the orientation of Ω produces an ellipse in PP.

Figure 5: Intersecting circles. If a circle with centre A has a point X inside, as well as a point Y outside, a second circle with centre B, and if X and Y lie on AB, then the circles have at least one common point on each side of AB.

Chord in a circle

Pascal's projective theoremThe 17th-century French mathematician Blaise Pascal proved that the three points (x, y, z) formed by intersecting the six lines that connect any six distinct points (A, B, C, D, E, F) on a circle are collinear.

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Currency. Money. Cash. Dollars. Bills. Pile of ten, twenty, fifty, and hundred dollar bills.

6 Questions About the History of Money and Banking Answered

Marriage. A couple getting married during a church wedding ceremony. (religion, ceremony, bride, groom)

The Seven Sacraments of the Roman Catholic church

Close up of books. Stack of books, pile of books, literature, reading. Homepage 2010, arts and entertainment, history and society

12 Novels Considered the “Greatest Book Ever Written”

Cheetah (Acinonyx jubatus) standing on rock, side view, Masai Mara National Reserve, Kenya

The Fastest Animals on Earth

illustration of the walking titanosaurus, Patagotitan mayorum

Titanosaurs: 8 of the World's Biggest Dinosaurs

Two chicks near an egg with a white background (poultry, chick, chickens, birds).

What Do Eggs Have to Do with Easter?

Nazi Germany, Nazi SS troops marching with victory standards at the Party Day rally in Nuremberg, Germany, 1933. (Schutzstaffel, Nazi Party, Nurnberg)

Why Was Nazi Germany Called the Third Reich?

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Photos and Videos

Eccentricity of conic sectionsThe eccentricity of a conic section completely characterizes its shape. For example, all circles have zero eccentricity, and all parabolas have unit eccentricity; hence, all circles (and all parabolas) have the same shape, only varying in size. (Under appropriate magnification they are indistinguishable.) In contrast, ellipses and hyperbolas vary greatly in shape.

The transformation of a circular region into an approximately rectangular regionThis suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

Figure 11: Beltrami's circle model.

concentric circles and infinity

Projective conic sectionsThe conic sections (ellipse, parabola, and hyperbola) can be generated by projecting the circle formed by the intersection of a cone with a plane (the reality plane, or RP) perpendicular to the cone's central axis. The image of the circle is projected onto a plane (the projective plane, or PP) that is oriented at the same angle as the cutting plane (Ω) passing through the apex (“eye”) of the double cone. In this example, the orientation of Ω produces an ellipse in PP.

Figure 5: Intersecting circles. If a circle with centre A has a point X inside, as well as a point Y outside, a second circle with centre B, and if X and Y lie on AB, then the circles have at least one common point on each side of AB.

Chord in a circle

Pascal's projective theoremThe 17th-century French mathematician Blaise Pascal proved that the three points (x, y, z) formed by intersecting the six lines that connect any six distinct points (A, B, C, D, E, F) on a circle are collinear.

Figure 10: Poincaré's circle model.

Understand how to calculate a circle's area with the help of chain and ruler

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Topics

circumference arc great circle diameter radius pi conic section parabola ellipse hyperbola

Key People

John Machin

English mathematician

Quiz

Equations written on blackboard

Numbers and Mathematics

Italian-born physicist Dr. Enrico Fermi draws a diagram at a blackboard with mathematical equations. circa 1950.

Define It: Math Terms