Equation of a Circle infographic - Standard Form Circle Equation

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Geometry

Equation of a Circle

Standard Form Circle Equation

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A circle in coordinate geometry is the set of all points that are the same distance from one fixed point called the center. Its equation lets you describe that shape exactly using algebra. This matters because circles appear in physics, engineering, computer graphics, and many geometry problems involving distance and symmetry.

The standard equation of a circle connects its center and radius directly to coordinates on the plane. If the center is (h,k)(h, k) and the radius is rr, then every point (x,y)(x, y) on the circle satisfies (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2. By reading or rewriting an equation, you can graph a circle, find its center and radius, and check whether a point lies on, inside, or outside the circle.

Key Facts

  • Standard form of a circle with center (h,k)(h, k) and radius rr: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2
  • If the center is at the origin, the equation becomes x2+y2=r2x^2 + y^2 = r^2
  • Radius is the distance from the center to any point on the circle
  • A point (x,y)(x, y) is on the circle if (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2
  • A point is inside the circle if (x−h)2+(y−k)2<r2(x - h)^2 + (y - k)^2 < r^2, and outside if (x−h)2+(y−k)2>r2(x - h)^2 + (y - k)^2 > r^2
  • From x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F = 0, complete the square to rewrite the equation in standard form

Vocabulary

Center
The fixed point (h, k) that is the same distance from every point on the circle.
Radius
The distance from the center of a circle to any point on the circle.
Standard form
The equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, which shows the center and radius directly.
Coordinate plane
A grid formed by the x-axis and y-axis used to locate points with ordered pairs.
Complete the square
An algebra method used to rewrite a quadratic expression so a circle equation can be put into standard form.

Common Mistakes to Avoid

  • Using the wrong signs for the center, because in (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 the center is (h,k)(h, k), so x+3x + 3 means h=−3h = -3, not 33.
  • Confusing rr with r2r^2, because the number on the right side of the equation is the radius squared, so if r2=25r^2 = 25 then the radius is 55.
  • Forgetting to square the radius term, because writing (x−h)2+(y−k)2=r(x - h)^2 + (y - k)^2 = r instead of r2r^2 gives the wrong circle.
  • Stopping before completing the square fully, because an equation like x2+y2+4x−6y−12=0x^2 + y^2 + 4x - 6y - 12 = 0 must be rewritten carefully to identify the correct center and radius.

Practice Questions

  1. 1 Write the equation of a circle with center (2, -3) and radius 4.
  2. 2 Find the center and radius of the circle x2+y2−6x+8y+9=0x^2 + y^2 - 6x + 8y + 9 = 0.
  3. 3 A point PP makes the left side of (x−1)2+(y+2)2=16(x - 1)^2 + (y + 2)^2 = 16 equal to 2020. Is PP on the circle, inside it, or outside it? Explain.