Standard Form of a Circle
Standard Form of a Circle: this Effortless Math guide explains the topic in plain language, shows how it works through solved examples, and gives you free practice to try immediately, so the idea sticks instead of staying abstract.
The equation of a circle is written using the radius and center of the circle.
Standard Form of a Circle: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Match the formIdentify the conic by its equation pattern.
- Read featuresFind the center, vertex, radius, axes, foci, or asymptotes.
- Sketch from anchorsPlot key points first, then draw the curve.
Worked examples
Circle center and radius
- Compare to circle standard form.
- The center is (4, -1).
- The radius is the square root of 25.
Parabola direction
- The x part is squared.
- The parabola opens up or down.
- The positive coefficient means it opens up.
Try one before moving on
Standard Form of a Circle: pop-up practice
The equation of the circle is shown with the center and radius of the circle. With this information, we can sketch the graph of the circle.
Related Topics
Step by Step Guide to Write the Standard Form of a Circle
- The Standard form of a circle is ((x- h)^2+( y-k)^2= r^2), where (r) is the radius and ((h, k)) is the center. By knowing the center and radius of the circle we can write the standard form of a circle.
Standard form of a Circle – Example 1:
Write the standard form equation of circle with center: ((0, 5)), radius: (3)
Solution:
The Standard form of a circle is ((x- h)^2+( y-k)^2= r^2), where (r) is the radius and ((h, k)) is the center.
In this case, the center is ((0, 5)) and the radius is (3): ((h, k)=(0, 5), r=3)
Then: ((x- 0)^2+( y-5)^2= 3^2 → x^2+( y-5)^2= 9 )
Standard form of a Circle – Example 2:
Write the standard form equation of the circle (x^2+y^2-6x+2y= 6).
Solution:
The Standard form of a circle is ((x- h)^2+( y-k)^2= r^2), where (r) is the radius and ((h, k)) is the center.
Group (x)-variables and (y)-variables together: ((x^2-6x)+( y^2+2y)= 6)
Convert (x) to square form: ((x^2-6x+9)+( y^2+2y)= 6+9 → (x-3)^2+( y^2+2y)=6+9)
Convert (y) to square form: ((x-3)^2+( y^2+2y+1)= 6+9+1 → (x-3)^2+(y+1)^2=6+9+1)
Then: ((x-3)^2+(y+1)^2=4^2)
Exercises for Writing Standard form of a Circle
Write the standard form equation of each circle with the given information.
- (color{blue}{Center: (0, 4)}, color{blue}{Radius: 2})
- (color{blue}{Center: (-1, 2)}), (color{blue}{Radius: 5})
- (color{blue}{x^2+y^2-6x+8y=0})
- (color{blue}{x^2+y^2-2x+8y=0})
- (color{blue}{x^2+(y-4)^2=2^2})
- (color{blue}{(x+1)^2+(y-2)^2=5^2})
- (color{blue}{(x-5)^2+y^2=4^2})
- (color{blue}{(x-1)^2+(y+4)^2=5^2})
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