Two Circles: Position and Common Tangents

Words you need first

A circle has a centre and a radius. For two circles we call the radii r1 and r2. The line joining the two centres is the line of centres, and its length is d, the distance between the centres.

A tangent to a circle touches it at exactly one point. A common tangent is one straight line that touches both circles. A common point is a point that lies on both circles.

All of the facts below depend on only three numbers: r1, r2 and d.

The five positions of two circles

Slide one circle towards the other and compare d with the sum (r1 + r2) and the difference (r1 − r2, taken as positive).

• Apart: d > r1 + r2. No common point. 4 common tangents.
• Touching outside (external tangency): d = r1 + r2. One common point. 3 common tangents.
• Crossing: difference < d < sum. Two common points. 2 common tangents.
• Touching inside (internal tangency): d = difference. One common point. 1 common tangent.
• One inside the other: d < difference. No common point. 0 common tangents.

When circles touch, the point of contact lies on the line of centres. For outside touching it is between the centres; for inside touching it is on the same line, beyond the smaller circle.

Common tangents and their lengths

A direct (outer) common tangent keeps both circles on the same side. A transverse (crossing) common tangent has one circle on each side and cuts the line of centres between the circles.

Draw the radii to the two touching points. They are both at 90° to the tangent, so they are parallel. Join the centres and drop a line to make a right-angled triangle. Pythagoras gives:

Direct tangent length = √(d² − (r1 − r2)²)

Transverse tangent length = √(d² − (r1 + r2)²)

The transverse tangent exists only when d ≥ r1 + r2, which is exactly when the circles are apart or touching outside. The direct one exists when d ≥ |r1 − r2|.

Try it: two coins on paper

Take two coins of different sizes and a ruler. Put the big coin down and draw round it. Slide the small coin until it just touches the big one outside: measure d between the centres and check d = r1 + r2. Now put the small coin on top so that it just touches from inside, near the edge: check d = r1 − r2. Lay the ruler along two coins on a table: that is an outer common tangent!

Key formulas and definitions

• Apart: d > r1 + r2 (4 common tangents)
• Touching outside: d = r1 + r2 (3 common tangents, 1 common point)
• Crossing: |r1 − r2| < d < r1 + r2 (2 common tangents, 2 common points)
• Touching inside: d = |r1 − r2| (1 common tangent, 1 common point)
• One inside the other: d < |r1 − r2| (0 common tangents)
• Length of direct (outer) common tangent = √(d² − (r1 − r2)²)
• Length of transverse (crossing) common tangent = √(d² − (r1 + r2)²)

Worked examples

Common mistakes

• Writing d = r1 − r2 for outside touching. Outside touching uses the sum r1 + r2; inside touching uses the difference.
• Forgetting that touching circles still have common tangents: 3 for outside touching, 1 for inside touching.
• Using the sum in the direct tangent formula. Direct uses (r1 − r2); transverse uses (r1 + r2).
• Mixing up "crossing" with "touching". Crossing circles share two points, touching circles share exactly one.