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Equation of a Circle

Standard form equation of a Circle

Equation of Circle Applet
applet
Circle Formulas
Circles
Circle Worksheets
worksheets
The standard form equation of a circle is a way to express the definition of a circle on the coordinate plane.
  • On the coordinate plane, the formula becomes $$(x -h)^2 + (y - k)^2 =r^2 $$
    • h and k are the x and y coordinates of the center of the circle
    • $$(x-9)^2 + (y-6)^2 =100 $$ is a circle centered at (9,6) with a radius of 10

Examples

General Formula
general formula equation of circle

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Circle with a center of (4,3) and a radius of 5
Example 1

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Circle with a center of (2, -1) and a radius of 4
example 2

Definition of Circle

Definition: A circle is the set of all points that are the same distance, r, from a fixed point.
General Formula: X 2 + Y 2=r2 where r is the radius

  • Unlike parabolas, circles ALWAYS have X 2 and Y 2 terms.
    • X2 + Y2=4 is a circle with a radius of 2 ( since 4 =22)
  • Remember that a circle is a locus of points. A circle is all of the points that are a fixed distance, known as the radius, from a given point, known as the center of the circle.
  • Explore the standard equation of a circle using the applet below (Go here for a larger version)
Circle Worksheets
worksheets
Circle Formulas
Circles

Equation of Circle Interactive HTML5 Applet

Applet on its own page
Explore and discover the standard form equation of a circle using the interactive circle below. To move the circle just click and drag on either of the two points, and the circle's standard form equation will adjust accordingly. You can also see the general form of the circle's equation by clicking 'show general form'
Radical Form? Show General Form? Equaton Type
$$ $$



Practice Problems


Picture of Equation of circle in standard form What is the equation of the circle pictured on the graph below?
Answer

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Look at the graph below, can you express the equation of the circle in standard form?
Picture of Equation of circle in standard form
Answer

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Look at the graph below, can you express the equation of the circle in standard form?
Answer
Picture of Equation of circle in standard form
What is the radius of the circles below?
Answer
  • Y2+X2 =9
  • Y2+X2 =16
  • Y2+X2 =25
  • Y2 + X2 = 11
  • Y2 + X2 = a
Look at each standard form equatio below and identify the center and radius.
Answer
  • (y-3)2+(x-1)2 =9
  • (y-5)2+(x-14)2 =16
  • (y-1)2+(x-5)2 =25
  • (x+2)2++(y-12)2 =36
  • (y+7)2+(x +5)2 =49
  • (x +8)2+(y+17)2 =49
Equation of Circle Applet
applet
Circle Formulas
Circles
Circle Worksheets
worksheets