Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point called the centre. The distance between any point on the circle and the centre is known as the radius. A line segment connecting two points on the circle while passing through the centre is called the diameter. This diameter represents the largest possible distance between any two points on the circle. The length of the diameter is exactly twice the length of the radius. A circle bounds a region of the plane called a disc. In strict mathematical usage, a circle refers only to the boundary of the disc. Everyday language often uses the term circle to refer to the entire disc area instead. An annulus describes a ring-shaped object bounded by two concentric circles. Any connected part of a circle is defined as an arc. Specifying two end points of an arc along with a centre allows for two arcs that together make up a full circle. A chord is a line segment whose endpoints lie on the circle itself. Chords divide a circle into two segments. A secant is an extended chord or a coplanar straight line intersecting a circle in two points. A tangent is a coplanar straight line that has one single point in common with the circle. It touches the circle at this specific point.
The circle has been known since before the beginning of recorded history. Prehistoric people made stone circles and timber circles long before written records existed. Circular elements are common in petroglyphs and cave paintings found across ancient landscapes. Disc-shaped prehistoric artifacts include the Nebra sky disc and jade discs called Bi. The Egyptian Rhind papyrus dated to 1700 BCE gives a method to find the area of a circle. This result corresponds to approximately 3.16049 as an approximate value of pi. Book 3 of Euclid's Elements deals with the properties of circles. Euclid provided a formal definition stating a circle is formed by lines drawn from a center. Plato explained the perfect circle in his Seventh Letter as different from any drawing or words. Early science particularly geometry and astrology was connected to the divine for most medieval scholars. Many believed there was something intrinsically divine or perfect that could be found in circles. In 1880 CE Ferdinand von Lindemann proved that pi is transcendental. This proof demonstrated that squaring the circle cannot be performed with straightedge and compass. With the advent of abstract art in the early 20th century geometric objects became an artistic subject. Wassily Kandinsky often used circles as an element of his compositions.
The ratio of a circle's circumference to its diameter is pi an irrational constant approximately equal to 3.141592654. The circumference C is related to the radius r and diameter d by specific formulas. As proved by Archimedes in his Measurement of a Circle the area enclosed equals that of a triangle. That triangle has a base length equal to the circle's circumference and height equal to the radius. A Cartesian coordinate system defines the circle with centre coordinates a comma b and radius r. This equation follows from the Pythagorean theorem applied to any point on the circle. If the circle is centred at the origin zero comma zero the equation simplifies significantly. The circle can also be written in parametric form using trigonometric functions sine and cosine. Here t acts as a parametric variable ranging from zero to two interpreted as an angle. An alternative parameterisation uses the tangent half-angle substitution but requires t to range through all reals. In polar coordinates the equation involves the distance from the origin to the centre and an anticlockwise angle. When the origin lies on the circle the equation becomes simpler still. In the complex plane a circle with a centre at c and radius r has a distinct equation. A slightly generalized equation for real p q and complex g is sometimes called a generalised circle.
Squaring the circle is the problem proposed by ancient geometers of constructing a square with the same area as a given circle. This construction must use only a finite number of steps with compass and straightedge. In 1882 the task was proven to be impossible as a consequence of the Lindemann-Weierstrass theorem. This theorem proves that pi is a transcendental number rather than an algebraic irrational number. It means pi is not the root of any polynomial with rational coefficients. Despite this impossibility the topic continues to be of interest for pseudomath enthusiasts. The proof relies on showing that no sequence of geometric operations can produce the required square area. Ancient geometers spent centuries attempting to solve this puzzle without success. Ferdinand von Lindemann's work in 1880 CE finally closed the door on these attempts. The mathematical community now accepts that squaring the circle cannot be done under strict Euclidean rules.
Natural circles are common such as the full moon or a slice of round fruit. The circle is the basis for the wheel which makes much of modern machinery possible. Related inventions such as gears rely on circular motion to function effectively. Astronomy and calculus were inspired by the study of the circle throughout history. The circle serves as the one-dimensional hypersphere known as the 1-sphere in topology. A curve of constant width has the same perpendicular distance between parallel lines regardless of direction. The circle is the simplest example of this type of figure. Taxicab geometry defines circles differently resulting in squares oriented at 45 degrees to coordinate axes. These shapes have a circumference equal to eight times their radius in that specific metric system. Topological definitions allow circles to transform into other forms via ambient isotopy. Regular polygons with n sides approach the circle as n approaches infinity. Archimedes applied this fact to approximate pi using inscribed and circumscribed polygons.
From the time of earliest known civilisations like the Assyrians and ancient Egyptians the circle has been used directly or indirectly in visual art. Artists along the Yellow River in China and Western civilisations of ancient Greece and Rome used it to convey messages. Differences in worldview had a great impact on artists' perceptions of the shape. Some emphasised the circle's perimeter to demonstrate democratic manifestation while others focused on its centre. This focus symbolises the concept of cosmic unity found in many traditions. In mystical doctrines the circle mainly symbolises the infinite and cyclical nature of existence. Religious traditions use it to represent heavenly bodies and divine spirits. The circle signifies sacred concepts including unity infinity wholeness the universe divinity balance stability and perfection. Such concepts have been conveyed through symbols like a compass a halo and the vesica piscis. Magic circles are part of some traditions of Western esotericism. A 13th-century manuscript shows the compass as a symbol of God's act of Creation. Notice also the circular shape of the halo surrounding figures in these religious depictions.
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Common questions
What is the definition of a circle in Euclidean geometry?
A circle is a shape consisting of all points in a plane that are at a given distance from a given point called the centre. The distance between any point on the circle and the centre is known as the radius.
When was it proven that squaring the circle is impossible?
In 1882 CE Ferdinand von Lindemann proved that pi is transcendental which demonstrated that squaring the circle cannot be performed with straightedge and compass. This proof relied on the Lindemann-Weierstrass theorem to show that no sequence of geometric operations can produce the required square area.
How does Archimedes describe the area of a circle?
Archimedes proved in his Measurement of a Circle that the area enclosed equals that of a triangle with a base length equal to the circle's circumference and height equal to the radius. He applied this fact to approximate pi using inscribed and circumscribed polygons.
Why do religious traditions use the circle symbol?
Religious traditions use the circle to represent heavenly bodies divine spirits sacred concepts including unity infinity wholeness the universe divinity balance stability and perfection. A 13th-century manuscript shows the compass as a symbol of God's act of Creation while halos surrounding figures depict circular shapes.