Circle-Circle Intersection — Definition, Formula & Examples

Circle-circle intersection refers to the point or points where two circles cross each other. Depending on the circles' sizes and the distance between their centers, they can intersect at zero, one, or two points.

Given two circles with centers $C_1$ , $C_2$ and radii $r_1$ , $r_2$ , let $d = |C_1 C_2|$ be the distance between centers. The circles intersect at two points if $|r_1 - r_2| < d < r_1 + r_2$ , at exactly one point (tangency) if $d = r_1 + r_2$ or $d = |r_1 - r_2|$ , and at no points if $d > r_1 + r_2$ or $d < |r_1 - r_2|$ .

Key Formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Where:

$d$ = Distance between the two circle centers
$(x_1, y_1)$ = Center of the first circle
$(x_2, y_2)$ = Center of the second circle

How It Works

To find intersection points algebraically, write each circle's equation in standard form and subtract one from the other. This eliminates the squared terms and produces a linear equation — the equation of the radical axis (a line). Solve this linear equation simultaneously with either circle's equation to find the coordinates of the intersection points. The number of solutions matches the number of intersection points predicted by the distance condition.

Worked Example

Problem: Find the intersection points of the circles

x^2 + y^2 = 25

and

(x-4)^2 + y^2 = 9

Step 1: Expand the second equation and subtract the first from it to get the radical axis equation.

x^2 - 8x + 16 + y^2 = 9 \;\Rightarrow\; \text{subtract } (x^2 + y^2 = 25):\; -8x + 16 = -16 \;\Rightarrow\; x = 4

Step 2: Substitute

x = 4

into the first circle's equation to find

y

16 + y^2 = 25 \;\Rightarrow\; y^2 = 9 \;\Rightarrow\; y = \pm 3

Step 3: Verify: the distance between centers is

d = 4

, and

|5 - 3| < 4 < 5 + 3

, confirming two intersection points.

Answer: The two circles intersect at

(4, 3)

and

(4, -3)

Why It Matters

Circle-circle intersection appears in coordinate geometry problems on the SAT and ACT, and in competition math. It is also essential in engineering and computer graphics, for example when determining the overlap of radar ranges or designing Venn diagrams with precise geometric boundaries.

Common Mistakes

Mistake: Trying to solve both circle equations independently without subtracting one from the other, leading to messy quartic expressions.

Correction: Always subtract one circle equation from the other first. This produces a linear equation (the radical axis), which greatly simplifies the algebra.