Circle
A circle is defined as the set of all points in a plane that are equidistant from a fixed point known as the center. It is a fundamental geometric shape and falls under the category of conic sections, formed by the intersection of a double-cone with a plane perpendicular to the cone. Key elements of a circle include the radius, which is the distance from the center to any point on the circle, and the diameter, which is a line segment passing through the center connecting two points on the circle. The circumference is the distance around the circle, and it maintains a constant ratio to the diameter, known as π (pi), an irrational number approximately equal to 3.14159.
In addition to its geometric properties, circles have significant applications in data representation, notably through pie charts, which visually display proportions of a whole. The mathematical representation of a circle can be expressed in Cartesian coordinates, and it plays an essential role in trigonometry, particularly through the unit circle, which simplifies the study of angles and periodic functions. Furthermore, circles extend into three dimensions as spheres, which have unique properties related to volume and surface area. Understanding circles is foundational in mathematics and has historical significance, with various cultures contributing to the study of π throughout history.
Circle
The circle consists of all points in a plane that are the same distance from a fixed point (called the "center of the circle"). All points in 3-dimensional space that are the same distance from a point form a "sphere," which is the 3-dimensional analog of a circle.
Overview
The circle is one of the "conics," which are formed by the intersection of a double-cone and a plane. The circle results when the plane is perpendicular to the cone.
The line segment from any point on a circle to the center is called a radius (plural radii). The term radius also refers to the length of this segment. Two radii mark out a sector of a circle. The line segment from any point on a circle through the center to the opposite side is called a diameter. The term diameter also refers to the length of this segment. A line segment joining any two points on a circle is a chord. A chord that goes through the center of the circle is a diameter.
The outside of the circle, also the distance around a circle, is the circumference. For any circle, the ratio of the circumference to the diameter has the same value. This value is represented by the Greek letter π, spelled in English as "pi" and pronounced as "pie." The ratio of the circumference to the radius is 2π. π is an irrational number, which means it has an endless decimal expression. To 10 decimal places π = 3.1415926536.
Circle Graph
One way of plotting data is by means of a circle graph (also called a pie chart), which shows graphically what proportion or percentage of the total is taken up by each data point. Each data point is represented by a sector of the circle, where the angular measure of the sector is the same percentage of the circumference of the circle that the data point is of the total.
Equations of a circle
The equation in Cartesian coordinates of a circle with center at the origin is x2 + y2 = r2 where r is the radius of the circle. This equation can also be written as y = ± √(r2 – x2). If the circle has center at (a, b) and radius r, its equation becomes (x – a)2 + (y – b)2 = r2.
The area of a circle with radius r is πr2..The area of a sector of a circle, where θ is the angle in degrees formed by the two radii, is θ/360 πr2.
The volume of a sphere of radius r is 4/3 πr3 and the surface area is 4 πr2.The circle has the property that it encloses a larger area than any polygon with perimeter equal to the circle’s circumference. Similarly a sphere encloses a larger volume than any polyhedron with the same surface area. This is why soap bubbles are spherical. The soap film is elastic and tries to form the smallest surface area, while the air inside tries to expand to the largest volume possible.
The Unit Circle
In the Cartesian coordinate plane (x-y plane), a circle with center at the origin and radius = 1 is called the unit circle. Each point on the circle can be uniquely specified by the angle the radius to the point makes with the positive x-axis, the angle being measured counterclockwise. There are two ways of measuring the angle: by degrees, with 360 degrees to the full circle; and by radians. One radian is the distance covered by going from the x-axis in a counterclockwise direction for a distance equal to the radius (which in the unit circle equals 1). The 180° angle of a semicircle is π radians. A full circle (360°) is 2π radians. One radian = 180/π degrees, or approximately 57.296° and one degree is approximately .01475 radians.
Why use radians? Certain mathematical formulas take a simpler form when expressed in radians rather than in degrees, for example the sine and cosine formulas given below. Because each point of the circle is uniquely specified by the angle θ with the x-axis, the x and y coordinates of the point are functions of θ.The x co-ordinate is given by the cosine function, abbreviated cos θ. The y coordinate is given by the sine function, abbreviated sin θ.
The slope of the radius line is given by the tangent function (tan θ) tan ( = sin ( / cos ( (that is, the slope is the y-coordinate divided by the x coordinate). By the Pythagorean theorem, sin2 θ + cos2 θ = 1. The sine and cosine functions can be computed by the Taylor series from elementary calculus. If the angle θ is given in radians the formulas are
Inscribed Angle
Two chords intersecting at a point on the circumference of a circle define an inscribed angle (angle ACB in the Figure 4).
If the center O is in the inside of the inscribed angle ACB, then the measure of angle ACB is one-half the measure of the central angle AOB. This shows that the inscribed angle formed by the chords from A and B to a point D on the circumference of the circle, such that the center O is inside the angle, have the same measure.
Great Circle
A circle on the surface of a sphere, whose center is the same as the center of the sphere, is known as a great circle. Another definition of great circle is the intersection of a sphere and a plane that passes through the center of the sphere. A great circle divides the sphere into two equal hemispheres. On the earth the equator and any meridian (line of longitude) are great circles. Any two points on the sphere define a unique great circle.
On the surface of a sphere, a great circle is the shortest distance between two points on the sphere. For example, traveling west from Los Angeles to Tokyo, the distance is approximately 5,780 nautical miles. Traveling on a great circle, the distance is approximately 5,490 nautical miles.
Trigonometry
The sine and cosine functions can also be applied to triangles. Consider the triangle ABC inside a unit circle, with point A at the center and point C on the circle. The distance AC is the radius, which is 1 because this is a unit circle. AB is called the adjacent side because it is adjacent to the angle CAB; BC is the opposite side.
The Unit Circle in the Complex Plane
There are interesting results from placing the unit circle in the complex plane. The center of the circle is at 0 + 0i (the origin) where i is √(–1). Each point on the circle now becomes the complex number cos θ + i sin θ.
When two points on the circle, represented here by the angles θ and φ, are multiplied together, the result is another point on the circle, specifically the one represented by the angle θ + φ. This suggests that angle θ is the logarithm of cos θ + i sin θ, and indeed some elementary calculus shows that cos θ + i sin θ = eiθ where e = 2.7182818… is the base of the natural logarithms. This provides the definition of complex exponents.
The product of cos θ + i sin θ and cos φ + i sin φ is
showing the formulas for sine and cosine of the sum of two angles are
History of π
The history of π shows how mathematics is a worldwide activity. The Ahmes papyrus (also called the Rhind papyrus) from ancient Egypt (circa 1900 BC) uses the value 3.1605 for π. The ancient Babylonians used 3.125 for π. In China Lui Hiu (third century AD) and Zu Chongzhi (fifth century AD) both computed π. Zu Chongzhi found a value of 355/113 = 3.1415929, which is correct to 6 decimal places. In India Madhava (circa 1500 AD) not only computed π but derived many of the series formulas used in trigonometry. In Classical Greece, Archimedes (circa 287-212 BC) showed that π is between 3 10/71 (3.1408) and 3 1/7 (3.142857).
In the Renaissance it became a game to compute π to more and more decimal places. By 1873 William Shanks, using only pencil and paper, computed 707 decimal places (the first 527 being correct). A more impressive feat was by Johann Dase who in 1844 computed 200 decimal places entirely in his head, without using pencil and paper. This feat took him about 2 months of mental effort. In the late 1940’s computers took over this computation. The first computer calculation produced 2,000 decimal places and, as of 2014, thirteen trillion decimal places have been computed.
Ancient Greek mathematicians tried to "square the circle," that is, to construct a line segment of length π with compass and straight-edge. In 1882 Ferdinand von Lindemann proved that π is transcendental, meaning that it is not the solution of any algebraic equation, nor can it be constructed with compass and straight-edge.
Bibliography
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McKellar, Danica. Girls Get Curves. New York: Penguin, 2012. Print.
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