Circle
Key Formula
Standard form: (x−h)2+(y−k)2=r2
Where:
- (h,k) = The coordinates of the center of the circle
- r = The radius — the fixed distance from the center to any point on the circle
- (x,y) = Any point on the circle
Worked Example
Problem: A circle has its center at (3, 2) and a radius of 5. Write the equation of the circle and determine whether the point (6, 6) lies on it.
Step 1: Write the standard form equation using the center (h, k) = (3, 2) and radius r = 5.
(x−3)2+(y−2)2=52
Step 2: Simplify the right side.
(x−3)2+(y−2)2=25
Step 3: Substitute the point (6, 6) into the left side to check if it satisfies the equation.
(6−3)2+(6−2)2=32+42=9+16=25
Step 4: Compare with the right side. Since 25 = 25, the point lies exactly on the circle.
25=25✓
Answer: The equation is (x − 3)² + (y − 2)² = 25, and the point (6, 6) does lie on the circle.
Another Example
This example works in the reverse direction — starting from an expanded (general form) equation and converting to standard form by completing the square, a key algebra skill.
Problem: Find the center and radius of the circle given by the equation x² + y² − 8x + 6y + 9 = 0.
Step 1: Group x-terms and y-terms together, and move the constant to the right side.
(x2−8x)+(y2+6y)=−9
Step 2: Complete the square for the x-group. Half of −8 is −4, and (−4)² = 16. Add 16 to both sides.
(x2−8x+16)+(y2+6y)=−9+16
Step 3: Complete the square for the y-group. Half of 6 is 3, and 3² = 9. Add 9 to both sides.
(x2−8x+16)+(y2+6y+9)=−9+16+9
Step 4: Factor each perfect square trinomial and simplify the right side.
(x−4)2+(y+3)2=16
Step 5: Read off the center and radius from standard form. The center is (4, −3) and r² = 16, so r = 4.
Center =(4,−3),r=4
Answer: The center is (4, −3) and the radius is 4.
Frequently Asked Questions
What is the difference between a circle and a sphere?
A circle is a two-dimensional figure: all points at a fixed distance from a center in a plane. A sphere is the three-dimensional analogue: all points at a fixed distance from a center in space. The equation of a sphere adds a third variable: (x − h)² + (y − k)² + (z − l)² = r².
What is the general equation of a circle?
The general (expanded) form is x² + y² + Dx + Ey + F = 0, where D, E, and F are constants. You can convert this to standard form (x − h)² + (y − k)² = r² by completing the square for both x and y. The center is (−D/2, −E/2) and r² = (D/2)² + (E/2)² − F.
Is a circle a function?
No. A full circle fails the vertical line test because most x-values correspond to two y-values (one above and one below the center). However, you can split a circle into an upper and lower semicircle, each of which is a function.
Circle vs. Ellipse
| Circle | Ellipse | |
|---|---|---|
| Definition | All points equidistant from one center point | All points where the sum of distances to two foci is constant |
| Equation (standard form) | (x − h)² + (y − k)² = r² | (x − h)²/a² + (y − k)²/b² = 1 |
| Symmetry | Infinite lines of symmetry (every diameter) | Exactly two lines of symmetry (major and minor axes) |
| Special relationship | A circle is a special case of an ellipse where a = b | An ellipse has two distinct radii (semi-major a and semi-minor b) |
| Area | πr² | πab |
Why It Matters
Circles appear throughout mathematics, from coordinate geometry problems in algebra and precalculus to trigonometry (the unit circle defines sine and cosine). In physics, circular motion, orbits, and wave patterns all rely on circle equations. Understanding the standard form equation is also essential preparation for studying conic sections — parabolas, ellipses, and hyperbolas.
Common Mistakes
Mistake: Forgetting to square the radius in the equation. Students often write (x − h)² + (y − k)² = r instead of r².
Correction: The right side of the standard form equation is always r², not r. If the radius is 5, the right side is 25.
Mistake: Getting the sign of the center coordinates wrong. Seeing (x − 3)² + (y + 4)² = 9 and stating the center as (−3, 4).
Correction: The standard form uses subtraction: (x − h) and (y − k). So (x − 3) means h = 3, and (y + 4) is really (y − (−4)), meaning k = −4. The center is (3, −4).
Related Terms
- Radius of a Circle or Sphere — The fixed distance defining the circle's size
- Circumference — Perimeter of a circle: C = 2πr
- Area of a Circle — Area enclosed by the circle: A = πr²
- Locus — A circle is a locus of equidistant points
- Point — The center is a point; the circle is a set of points
- Diameter — A chord through the center; equals 2r
- Chord — A line segment with both endpoints on the circle
- Tangent Line — A line touching the circle at exactly one point