1. Keystone Geometry Unit 7
Equation of A Circle
Keystone Geometry
Unit 7

2. Equation of a Circle The center of a circle is given by (h, k).
The radius of a circle is given by r.
The equation of a circle in
standard form is:
(x – h)2 + (y – k)2 = r2

3. Finding the Equation of a Circle
Circle O
The center is (0, 0) so it
is located at the origin.
The radius is 12.
The equation is
x 2 + y 2 = 144

4. Finding the Equation of a Circle
Circle A
The center is (16, 10)
The radius is 10
The equation is
(x – 16)2 + (y – 10)2 = 100
Circle B
The center is (4, 20)
The radius is 10
(x – 4)2 + (y – 20)2 = 100

5. Graphing Circles
(x – 3)2 + (y – 2)2 = 9
Center (3, 2)
Radius of 3

6. Graphing Circles
(x + 4)2 + (y – 1)2 = 25
Center (-4, 1)
Radius of 5

7. Graphing Circles
(x – 5)2 + y2 = 36
Center (5, 0)
Radius of 6

8. Writing Equations of Circles
Write the standard equation of the circle with a center at (4, 7) and a radius of 5. (x – 4)2 + (y – 7)2 = 25 (-3, 8) and a radius of 6.2 (x + 3)2 + (y – 8)2 = 38.44

9. Writing Equations of Circles
Write the standard equation of the circle with a center at (2, -9) and a radius of . (x – 2)2 + (y + 9)2 = 11 (0, 6) and a radius of . x 2 + (y – 6)2 = 7

10. Writing Equations of Circles
Write the standard equation of the circle with a center at (-1.9, 8.7) and a radius of 3. (x + 1.9)2 + (y – 8.7)2 = 9

11. Identify the center and radius and sketch the graph:
Remember, the center values end up being the opposite sign of what is with the x and y and the right hand side is the radius squared. So the center is at (-4,3) and the radius is 5. 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8

12. But what if the equation of a circle is not in standard form but multiplied out (FOILED)!
Moving everything to one side in descending order and combining like terms we’d have:
If we'd have started with it like this, we'd have to complete the square on both the x's and y's to get in standard form.

13. Move constant to the other side
Group x terms and a place to complete the square
Group y terms and a place to complete the square 4 16 4 16
Complete the square, write factored form, and we would be back to the standard form for the equation of a circle with a center at (-2,4) and a radius of 2.

14. Writing a Standard Equation of a Circle
The point (1, 2) is on a circle whose center is (5, -1). Write a standard equation of the circle.
First solve for the radius:
Then plug h, k and r in to the standard equation of a circle:

15. Another example:
Find the center, the length of the radius, and write the equation of the circle if the endpoints of a diameter are (-8,2) and (2,0).
Center: Use midpoint formula!
Length: use distance formula with radius and an endpoint
Equation: Put it all together