Intersection of circles

If we have two circles and know their radii and the distance between their centres, we can find out how their circles overlap.

Ways they can overlap

There are 6 ways that two circles can overlap:

Completely separate (no points of intersection)
Intersecting at two points (the circles properly overlap, but neither is inside the other)
Internally tangential (the circles touch at just one point, from the inside, one circle is inside the other)
Externally tangiential (the circles touch at just one point, from the outside, one circle is not inside the other)
Touching at exactly one point (called tangent), of which they can be:
- Externally tangential (the circles touch from the outside, one circle is not inside the other)
- Internally tangential (the circles touch from the inside, one circle is inside the other)
One circle inside the other (the circles don’t touch, but one is completely inside the other)
The circles are the same (the circles have the same centre and radius, so they are literally the same shape)

Checking whether they overlap

If we know the radius of circle 1 and circle 2 (r_1 and r_2), and the distance between their centres (d), we can check which of the 6 cases applies by checking some inequalities.

Separate circles

Two circles are completely separate if the distance between their centres is greater than the sum of their radii:

d > r_1 + r_2

Intersecting circles

Two circles intersect at two points if the distance between their centres is less than the sum of their radii, but greater than the difference of their radii:

|r_1 - r_2| < d < r_1 + r_2

Internally tangential circles

Two circles are internally tangential if the distance between their centres is equal to the difference of their radii:

d = |r_1 - r_2|

Externally tangential circles

Two circles are externally tangential if the distance between their centres is equal to the sum of their radii:

d = r_1 + r_2

One circle inside the other

Two circles are such that one is completely inside the other if the distance between their centres is less than the difference of their radii:

d < |r_1 - r_2|

The circles are the same

Two circles are the same if the distance between their centres is zero, and their radii are the same:

d = 0 \quad \text{and} \quad r_1 = r_2

flashcards

Question	Answer
What is the condition for two circles to be completely separate?	d > r_1 + r_2
What is the condition for two circles to intersect at two points?	$
What does it mean if the distance between two circle centres equals the absolute difference of their radii?	The circles are internally tangential (they touch at one point from the inside).
What does it mean if the distance between two circle centres equals the sum of their radii?	The circles are externally tangential (they touch at one point from the outside).
What is the condition for one circle to be completely inside the other (without touching)?	$d <
How do you determine if two circles are the same circle?	d = 0 and r_1 = r_2
List the 6 ways two circles can overlap.	1. Completely separate 2. Intersecting at two points 3. Internally tangential 4. Externally tangential 5. One circle inside the other 6. The circles are the same