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Arc (geometry)

 

 
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Arc (geometry)
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, an arc is a closed
Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
 segment of a differentiable curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 in the two-dimensional plane; for example, a circular arc is a segment of the circumference
Circumference

The circumference is the distance around a closed curve. Circumference is a kind of perimeter....
 of a circle. If the arc segment occupies a great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
 (or great ellipse
Great ellipse

.A great ellipse is an ellipse passing through two point on a spheroid and having the same center as that of the spheroid. Equivalently,it is an ellipse on the surface of a cylinder centered at the origin ....
), it is considered a great-arc segment.

The length
Arc length

Determining the length of an irregular arc segment ? also called rectification of a curve ? was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form expression in some cases....
 of an arc of a circle with radius and subtending an angle (measured in radian
Radian

The radian is a unit of plane angle, equal to 180/pi Degree , or about 57.2958 degrees, or about 57?17'45?. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
s) with the circle center — i.e., the central angle
Central angle

A central angle is an angle whose vertex is the center of a circle, and whose sides pass through a pair of points on the circle, thereby subtending an Arc between those two points whose angle is equal to the central angle itself....
 — equals . This is because








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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, an arc is a closed
Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
 segment of a differentiable curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 in the two-dimensional plane; for example, a circular arc is a segment of the circumference
Circumference

The circumference is the distance around a closed curve. Circumference is a kind of perimeter....
 of a circle. If the arc segment occupies a great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
 (or great ellipse
Great ellipse

.A great ellipse is an ellipse passing through two point on a spheroid and having the same center as that of the spheroid. Equivalently,it is an ellipse on the surface of a cylinder centered at the origin ....
), it is considered a great-arc segment.

The length
Arc length

Determining the length of an irregular arc segment ? also called rectification of a curve ? was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form expression in some cases....
 of an arc of a circle with radius and subtending an angle (measured in radian
Radian

The radian is a unit of plane angle, equal to 180/pi Degree , or about 57.2958 degrees, or about 57?17'45?. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
s) with the circle center — i.e., the central angle
Central angle

A central angle is an angle whose vertex is the center of a circle, and whose sides pass through a pair of points on the circle, thereby subtending an Arc between those two points whose angle is equal to the central angle itself....
 — equals . This is because

Substituting in the circumference

and solving for arc length, , in terms of yields

An angle of degrees has a size in radians given by

and so the arc length equals

See also

  • Arc length
    Arc length

    Determining the length of an irregular arc segment ? also called rectification of a curve ? was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form expression in some cases....
  • Other meanings of arc
    Arc

    Arc may refer to:...
  • Circular-arc graph
    Circular-arc graph

    In graph theory, a circular-arc graph is the intersection graph of a set of Arc on the circle. It has one vertex for each arc in the set, and an edge between every pair of vertices corresponding to arcs that intersect....


External links

  • With interactive animation
  • Arcs, arc central angle, arc peripheral angle, central angle theorem and others.