Touching Circles

Circles can touch internally, externally, or not at all.

Internal Touching

Circles touch internally when the distance between centres equals the difference of the radii.

External Touching

Circles touch externally when the distance between centres equals the sum of the radii.

Not Touching

Circles may not touch at all if the distance between centres is too large or one circle lies completely inside the other.

Do They Touch?

1. Find the distance between the centres.
2. Add the radii — if equal, circles touch externally.
3. Subtract the radii — if equal, circles touch internally.

Interactive: Explore Touching Circles

Drag the red circle, adjust radii, and investigate internal/external tangency.

Example

Do the circles with equations:

and touch ?

The first circle has centre A(-3, 2) and radius:

The second circle has centre B(6, -1) and radius:

Using the distance formula:

Since AB = r₁ + r₂, the circles touch externally.

Do the circles touch?

Consider the circles with equations:

Using the distance formula:

Since AB = r₁ − r₂, the circles touch internally.

Using Points to Find Centres

Example

The point C(-1,4) is the point of contact...

Using the distance formula:

This is called the stepping out method.

The equation of the small circle is:

Using the Section Formula

Alternatively, using the section formula:

The equation of the small circle is: