# Article

# Article

- Mathematics
- Geometry
- Circle

# Circle

Article By:

**Blumenthal, Leonard M. **Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.

**Frame, J. Sutherland **Formerly, Department of Mathematics, Michigan State University, East Lansing, Michigan.

Last updated:2014

DOI:https://doi.org/10.1036/1097-8542.136800

**The curve that is the locus of points in a plane with equal distance (radius) from a fixed point (center) ( Fig. 1).** In elementary mathematics, circle often refers to the finite portion of the plane bounded by a curve (circumference) all points of which are equidistant from a fixed point of the plane, that is, a circular disk. Circles are conic sections and are defined analytically by certain second-degree equations in cartesian coordinates. They were extensively studied by the ancient Greeks, who formulated the famous problem of “squaring the circle,” that is, to construct, with compasses and unmarked straightedge only, a square whose area is equal to that of a given circle. It was not until 1882 that this was shown to be impossible, when F. Lindemann proved that the ratio of the length of a circle to its diameter (denoted by π) is not the root of any algebraic equation with integer coefficients. That π is irrational (that is, not the quotient of two integers) was shown by A. M. Legendre in 1794. Numerous approximations of π appeared quite early; for example, 3 (

*Book of Kings*); 3

^{1}/

_{7}> π > 3

^{10}/

_{71}(Archimedes); $\sqrt{\text{10}}$ (

*Ch'ang Höng*, a.d. 78–139); and 3.1416 (Aryabhata, a.d. 476–550). Electronic computers have calculated π to 10

^{13}decimal places. Interesting expressions are

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