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 Circle with a center of (4,3) and a radius of 5Advertisement
Another Example
Circle with a center of (2, -1) and a radius of 4Definition 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)
Practice Problems
Practice 1
What is the equation of the circle pictured on the graph below?
Since the radius of this this circle is 1, and its center is the origin, this picture's equation is
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Practice 2
Look at the graph below, can you express the equation of the circle in standard form?
Since the radius of this this circle is 1, and its center is (1,0) , this circle's equation is
Practice 3
Look at the graph below, can you express the equation of the circle in standard form?
Since the radius of this this circle is 2, and its center is (3,1) , this circle's equation is
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Practice 4
What is the radius of the circles below?
Y2+X2 = 9
Y2+X2 = 16
Y2+X2 = 25
Y2 + X2 = 11
Y2 + X2 = a
Practice 5
Look at each standard form equation below and identify the center and radius.
r = radius
(y-3)2+(x-1)2 = 9
(1, 3) r = 3
(y-5)2+(x-14)2 = 16
(14, 5) r = 4
(y-1)2+(x-5)2 = 25
(5, 1) r = 5
(x+2)2++(y-12)2 = 36
(-2, 12) r = 6
(y+7)2+(x +5)2 = 49
(-5, -7) r = 7
(x +8)2+(y+17)2 = 49
(-8, -17) r = 7
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