Difference between revisions of "Circle"
(username removed) (Added parametric equation for a circle) |
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[[Image:265px-Circle Area svg.png|right|thumb|Area of the circle = <math>\pi</math> × area of the shaded square]] | [[Image:265px-Circle Area svg.png|right|thumb|Area of the circle = <math>\pi</math> × area of the shaded square]] | ||
| − | A '''circle''' is the [[set]] of all the points in a given [[plane]] that are the same distance, called the [[radius]], ''r'', from a given [[point]] called the center. | + | A '''circle''' is the [[set]] of all the points in a given [[plane]] that are the same distance, called the [[radius]], ''r'', from a given [[point]] called the [[center]]. |
A circle may also be defined [[algebra]]ically, as the set of solutions to an equation of the form: | A circle may also be defined [[algebra]]ically, as the set of solutions to an equation of the form: | ||
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:<math>(x-a)^2 + (y-b)^2 = r^2</math>, | :<math>(x-a)^2 + (y-b)^2 = r^2</math>, | ||
where <math>(a,b)</math> is the center of the circle. | where <math>(a,b)</math> is the center of the circle. | ||
| + | It can also be described parametrically in terms of a parameter <math>t</math> as: | ||
| − | + | :<math>x = a + r \cos{t}</math> | |
| + | :<math>y = b + r \sin{t}</math> | ||
| − | :<math>C = 2\pi | + | The distance around the circle, or [[circumference]], is given by: |
| + | |||
| + | :<math>C = 2\pi r</math> | ||
The area inside the circle is calculated using the formula: | The area inside the circle is calculated using the formula: | ||
| − | :<math>A = \pi | + | :<math>A = \pi r^2</math> |
| + | |||
| + | This can be easily derived from the area of an [[ellipse]]. For an [[ellipse]] with major and minor axis <math>a</math> and <math>b</math> respectively, the area is <math>\pi ab</math>. Setting the major an minor axis equal to each other give the formula for a circle. | ||
A circle is a [[conic section]], the intersection of a plane with a cone such that the plane is perpendicular to the axis of the cone. | A circle is a [[conic section]], the intersection of a plane with a cone such that the plane is perpendicular to the axis of the cone. | ||
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Circles can readily be constructed by using a fixed distance between a pencil point and the center. A [[compass]] is a tool for doing this easily. | Circles can readily be constructed by using a fixed distance between a pencil point and the center. A [[compass]] is a tool for doing this easily. | ||
| − | A circle with <math>r = 1</math> is called the ''unit circle'', and is used extensively in [[ | + | A circle with <math>r = 1</math> is called the ''unit circle'', and is used extensively in [[trigonometry]]. |
| − | [[Category:Geometry | + | [[Category:Plane Geometry]] |
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Latest revision as of 17:46, November 8, 2016
A circle is the set of all the points in a given plane that are the same distance, called the radius, r, from a given point called the center.
A circle may also be defined algebraically, as the set of solutions to an equation of the form:
- <math>(x-a)^2 + (y-b)^2 = r^2</math>,
where <math>(a,b)</math> is the center of the circle. It can also be described parametrically in terms of a parameter <math>t</math> as:
- <math>x = a + r \cos{t}</math>
- <math>y = b + r \sin{t}</math>
The distance around the circle, or circumference, is given by:
- <math>C = 2\pi r</math>
The area inside the circle is calculated using the formula:
- <math>A = \pi r^2</math>
This can be easily derived from the area of an ellipse. For an ellipse with major and minor axis <math>a</math> and <math>b</math> respectively, the area is <math>\pi ab</math>. Setting the major an minor axis equal to each other give the formula for a circle.
A circle is a conic section, the intersection of a plane with a cone such that the plane is perpendicular to the axis of the cone.
Circles can readily be constructed by using a fixed distance between a pencil point and the center. A compass is a tool for doing this easily.
A circle with <math>r = 1</math> is called the unit circle, and is used extensively in trigonometry.