Parallel of latitude approximately 66°30' north of the Equator that circumscribes the northern frigid zone. It marks the southern limit of the area within which, for one day or more each year, the sun does not set or rise. The length of continuous day or night increases northward from the Arctic Circle, mounting to six months at the North Pole.
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Parallel of latitude approximately 66°30' south of the Equator that circumscribes the southern frigid zone. It marks the northern limit of the area within which, for one day or more each year, the sun does not set or rise. The length of continuous day or night increases southward from the Antarctic Circle, mounting to six months at the South Pole.
Learn more about Antarctic Circle with a free trial on Britannica.com.
Circles are simple shapes of Euclidean geometry consisting of those points in a plane which are at a constant distance, called the radius, from a fixed point, called the center. A circle with center A is sometimes denoted by the symbol .
A chord of a circle is a line segment whose both endpoints lie on the circle. A diameter is a chord passing through the center. The length of a diameter is twice the radius. A diameter is the largest chord in a circle.
Circles are simple closed curves which divide the plane into an interior and an exterior. The circumference of a circle is the perimeter of the circle, and the interior of the circle is called a disk. An arc is any connected part of a circle.
The circle has been known since before the beginning of recorded history. It is the basis for the wheel which, with related inventions such as gears, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. Some highlights in the history of the circle are:
In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that
It can be proven that a conic section is a circle if and only if the point I(1: i: 0) and J(1: −i: 0) lie on the conic section. These points are called the circular points at infinity.
In polar coordinates the equation of a circle is
In the complex plane, a circle with a center at c and radius (r) has the equation . Since , the slightly generalised equation for real p, q and complex g is sometimes called a generalised circle. Not all generalised circles are actually circles: a generalized circle is either a (true) circle or a line.
The tangent line through a point P on a circle is perpendicular to the diameter passing through P. The equation of the tangent line to a circle of radius r centered at the origin at the point (x1, y1) is
Hence, the slope of a circle at (x1, y1) is given by:
More generally, the slope at a point (x, y) on the circle , i.e., the circle centered at (a, b) with radius r units, is given by
provided that .
The numeric value of never changes.
In modern English, it is (as in apple pie).
Area = r^2 cdot pi
Using a square with side lengths equal to the diameter of the circle, then dividing the square into four squares with side lengths equal to the radius of the circle, take the area of the smaller square and multiply by .
that is, approximately 79% of the circumscribing square.
Another proof of this result which relies only on two chord properties given above is as follows. Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know it is part of a diameter of the circle. Since the diameter is twice the radius, the “missing” part of the diameter is (2r−x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (2r−x)x=(y/2)². Solving for r, we find the required result.
An inscribed angle is exactly half of the corresponding central angle (see Figure). Hence, all inscribed angles that subtend the same arc have the same value (cf. the blue and green angles in the Figure). Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle.
Apollonius of Perga showed that a circle may also be defined as the set of points in plane having a constant ratio of distances to two fixed foci, A and B. That circle is sometimes said to be drawn about two points.
The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:
Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to , the angle CPD is exactly , i.e., a right angle. The set of points P that form a right angle with a given line segment CD form a circle, of which CD is the diameter.