Area of a circle

Area of a diskarea enclosed by a circlearea inside a circlearea of the diskarea of this circlecircular areahis doubling methodr 2 specifying the area
In geometry, the area enclosed by a circle of radius r iswikipedia
97 Related Articles

Archimedes

Archimedes of SyracuseArchimedeanArchimedes Heat Ray
One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons. Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle.
Generally considered the greatest mathematician of antiquity and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola.

Circle

circularcircles360 degrees
In geometry, the area enclosed by a circle of radius r is
As proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to multiplied by the radius squared:

Integral

integrationintegral calculusdefinite integral
Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis.
This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle.

Area

surface areaArea (geometry)area formula
In geometry, the area enclosed by a circle of radius r is
(The circumference is 2r, and the area of a triangle is half the base times the height, yielding the area r 2 for the disk.) Archimedes approximated the value of π (and hence the area of a unit-radius circle) with his doubling method, in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular hexagon, then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with circumscribed polygons).

Disk (mathematics)

diskdiscdisks
Although often referred to as the area of a circle in informal contexts, strictly speaking the term disk refers to the interior of the circle, while circle is reserved for the boundary only, which is a curve and covers no area itself.
The area of a closed or open disk of radius R is πR 2 (see area of a disk).

Pi

ππ\pi
. Here the Greek letter constant, approximately equal to 3.14159, which is equal to the ratio of the circumference of any circle to its diameter.
.* The area of a circle with radius

Lune of Hippocrates

quadrature of the lunelunes of Alhazenlunes of Hippocrates
Prior to Archimedes, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates, but did not identify the constant of proportionality.
Heath concludes that, in proving his result, Hippocrates was also the first to prove that the area of a circle is proportional to the square of its diameter.

Isoperimetric inequality

isoperimetricisoperimetric problemisoperimetric inequalities
This is known as the isoperimetric inequality, which states that if a rectifiable Jordan curve in the Euclidean plane has perimeter C and encloses an area A (by the Jordan curve theorem) then
The area of a disk of radius R is πR 2 and the circumference of the circle is 2πR, so both sides of the inequality are equal to 4π 2 R 2 in this case.

Geometry

geometricgeometricalgeometries
In geometry, the area enclosed by a circle of radius r is

Radius

radiiradialradially
In geometry, the area enclosed by a circle of radius r is

Constant (mathematics)

constantconstantsConstant functions
. Here the Greek letter constant, approximately equal to 3.14159, which is equal to the ratio of the circumference of any circle to its diameter.

Circumference

circumferentialgirthcircumferential line
. Here the Greek letter constant, approximately equal to 3.14159, which is equal to the ratio of the circumference of any circle to its diameter.

Diameter

Ddiameters
. Here the Greek letter constant, approximately equal to 3.14159, which is equal to the ratio of the circumference of any circle to its diameter.

Limit (mathematics)

limitlimitsconverge
One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons.

Regular polygon

regularregular polygons30-gon
One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons.

Perimeter

Perimeter lengthperimeter of the polygon
The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and the corresponding formula (that the area is half the perimeter times the radius, i.e.

Apothem

distance from its center to its sides
The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and the corresponding formula (that the area is half the perimeter times the radius, i.e.

Curve

closed curvespace curvesmooth curve
Although often referred to as the area of a circle in informal contexts, strictly speaking the term disk refers to the interior of the circle, while circle is reserved for the boundary only, which is a curve and covers no area itself.

Real analysis

realtheory of functions of a real variablefunction theory
Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis.

Ancient Greece

Greekancient Greekancient Greeks
However the area of a disk was studied by the Ancient Greeks.

Eudoxus of Cnidus

EudoxusEudoxus of CnidosEudoxan planetary model
Eudoxus of Cnidus in the fifth century B.C. had found that the area of a disk is proportional to its radius squared.

Euclidean geometry

plane geometryEuclideanEuclidean plane geometry
Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle.

Right triangle

right-angled triangleright angled triangleright angle triangle
Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle.

Measurement of a Circle

On the Measurement of the CircleMeasurement of the Circle
Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle.

Hippocrates of Chios

Hippocrates
Prior to Archimedes, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates, but did not identify the constant of proportionality.