Equation of a Circle

Standard form equation of a Circle

This page: Angle formed by intersecting chords | Tangent and Chord||The perpendicular Bisector of a chord| Chords Equidistant from Center | product segments of intersecting chords
Related Pages: Circles forumula, graph, equations | Equation of A Circle | Circumference | Area | Chord |Arc | Inscribed Angles|Secant of circle | 2 Tangents from 1 point | Tangents, Secants, Circles Theorems

The standard form equation of a circle is a way to express the definition of a circle on the coordinate plane.

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.
On the coordinate plane, the formula becomes (X-H)² + (Y-K)²=r²

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

Definition 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=r²where 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)

On the coordinate plane, the formula is similar.

Interactive Circle

Interactive equation of circle activity. Click and drag either of the points below to explore the standard form equation of a circle and its relationship to the radius.

Practice Problems

What is the equation of the circle pictured on the graph below?
Picture of Equation of circle in standard form

Answer

Look at the graph below, can you express the equation of the circle in standard form?
Picture of Equation of circle in standard form

Answer

Look at the graph below, can you express the equation of the circle in standard form?

Answer
Picture of Equation of circle in standard form

Y²+X² =9
Y²+X² =16
Y²+X² =25
Y²+X² =36
Y²+X² =49

What is the radius of the circles on the left?

Radius of Circles

Equation of Circle in Standard Form

(y-3)²+(X-1)² =9
(y-5)²+(X-14)² =16
(y-1)²+(X-5)² =25
(X+2)²⁺+(y-12)² =36
(y+7)²+(X +5)² =49
(X +8)²+(y+17)² =49

What is the center and radius of each circle to the left?

Answers

(1, 3) r = 3
(14, 5) r = 4
(5, 1) r = 5
(-2, 12) r = 6
(-5, -7) r = 7
(-8, -17) r = 7

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