What Does It Mean for Two Circles to Touch?
Two circles touch each other (are tangent to each other) when they meet at exactly one point, called the point of contact. If they met at two points they would be intersecting; if they met at none they would be separate or nested. One shared point is the borderline case, and it comes in two flavours: the circles can touch from outside each other or from inside.
The single most useful fact is that the point of contact and the two centres are collinear, meaning all three lie on one straight line, the line of centres. This is why the whole problem reduces to a single distance measurement between the two centres $O_1$ and $O_2$. The circles themselves build on the basic idea of circles and their radius.
By the end you will be able to look at two radii and one centre-distance and say instantly whether the circles touch, and how.
When one coin resting against another shares a single point
Slide two coins across a table until their rims just meet, and for one instant they share a single point of contact, no more. That single-point meeting is the whole idea behind two circles touching each other, and it splits cleanly into two cases with two neat distance rules.
How Do You Tell If Circles Touch Externally or Internally?
Measure the distance $d$ between the two centres and compare it with the radii. That single comparison decides everything.
External touch: the circles sit outside each other and meet at one point. This happens exactly when $d = R + r$.
Internal touch: the smaller circle sits inside the larger and touches its inner edge at one point. This happens exactly when $d = |R - r|$ (the difference of the radii).
Here $R$ and $r$ are the two radii and $d$ is the centre-to-centre distance. The reason both formulas work traces straight back to the collinear-centres fact: walking along the line of centres, you cover one radius, reach the contact point, then either add the second radius (going outward) or subtract it (staying inside).
Do two circles touch if the distance between centres equals the sum of the radii? Yes, that is the exact external-tangency condition. If $d$ were larger than $R + r$ the circles would be fully separate; if smaller, they would cut each other at two points.
A quick summary of every configuration by distance:
Distance between centres | Relationship | Common points |
|---|---|---|
$d > R + r$ | Separate (outside) | 0 |
$d = R + r$ | Touch externally | 1 |
$\lvert R - r\rvert < d < R + r$ | Intersecting | 2 |
$d = \lvert R - r\rvert$ | Touch internally | 1 |
$d < \lvert R - r\rvert$ | One inside the other | 0 |
When $d = 0$ the two centres coincide and you get concentric circles, which never touch at all.
What Are the Properties of Touching Circles?
Touching circles share a small set of properties that make problems fast to solve once you spot them.
Collinear centres and contact point. The two centres $O_1$, $O_2$ and the point of contact $P$ always lie on one straight line. Any problem that gives you two of these hands you the third.
A shared tangent at the contact point. At $P$, both circles have the same tangent line, perpendicular to the line of centres. This common tangent separates the two circles when they touch externally and rests alongside them when they touch internally.
The contact point divides the line of centres. For external touch, $P$ lies between $O_1$ and $O_2$; for internal touch, $P$ lies beyond the smaller centre, on the far side.
Distance is decisive. Because the relationship is fixed entirely by $d$ versus $R \pm r$, no angle or coordinate is needed to classify the contact.
These sit inside the wider family of circle theorems that govern chords, angles, and tangents.
How Many Common Tangents Do Touching Circles Have?
The count of shared tangent lines is a favourite exam question, and touching circles are the crossover cases.
External touch: three common tangents, two running alongside the pair plus one passing through the contact point.
Internal touch: one common tangent, the shared line at the contact point.
Separate circles have four common tangents and intersecting circles have two, so the touching cases are exactly where the count drops from four to two. The full picture is worked out in common tangents, and the equal-length result for two tangents drawn from one outside point is covered in tangents from an external point.
Examples of Circles Touching Each Other
Example 1
Two circles have radii $5$ cm and $3$ cm, and the distance between their centres is $8$ cm. Do they touch, and how?
Compare $d$ with $R + r$:
$$R + r = 5 + 3 = 8 \text{ cm}$$
Since $d = 8 = R + r$, the circles touch externally.
Example 2
Two circles have radii $6$ cm and $2$ cm with centres $4$ cm apart. A student computes $R + r = 8$, sees $4 \neq 8$, and concludes the circles do not touch. Where is the slip?
The student checked only the external condition. Checking just $d = R + r$ misses the internal case entirely, and internal touch is exactly what a small centre-distance signals.
Test the internal condition instead:
$$\lvert R - r\rvert = \lvert 6 - 2\rvert = 4 \text{ cm}$$
Since $d = 4 = |R - r|$, the circles touch internally. Always compare $d$ against both $R + r$ and $|R - r|$ before deciding.
Example 3
Circle A has radius $7$ cm and circle B touches it externally with centres $11$ cm apart. Find the radius of circle B.
External touch means $d = R + r$:
$$11 = 7 + r$$
$$r = 11 - 7 = 4 \text{ cm}$$
The radius of circle B is $4$ cm.
Example 4
Two circles of radii $9$ cm and $4$ cm touch internally. Find the distance between their centres.
Internal touch means $d = |R - r|$:
$$d = \lvert 9 - 4\rvert = 5 \text{ cm}$$
The centres are $5$ cm apart.
Example 5
A circle of radius $10$ cm has a smaller circle touching it internally, and the centres are $6$ cm apart. Find the smaller radius.
Internal touch gives $d = R - r$ (with $R$ the larger):
$$6 = 10 - r$$
$$r = 10 - 6 = 4 \text{ cm}$$
The smaller radius is $4$ cm.
Example 6
Circle $C_1: x^2 + y^2 = 4$ and circle $C_2: (x-5)^2 + y^2 = 9$. Do they touch?
Read off the centres and radii: $C_1$ has centre $(0,0)$, radius $R = 2$; $C_2$ has centre $(5,0)$, radius $r = 3$. The centre-distance is:
$$d = \sqrt{(5-0)^2 + (0-0)^2} = 5$$
Compare: $R + r = 2 + 3 = 5$. Since $d = 5 = R + r$, the circles touch externally. This coordinate version connects to the equation of a circle.
Where Do Touching Circles Show Up?
The touching condition matters wherever round parts sit against each other and one wrong distance ruins the fit.
Gears and bearings. Two meshing gear pitch-circles are designed to touch externally; if the centre-distance drifts from $R + r$, the teeth bind or slip. Ball bearings ride internally against a race, a textbook internal-tangency layout.
Packing and coin problems. Arranging coins, pipes, or cables so each rests against its neighbours is a chain of external-tangency conditions.
Coordinate geometry and design software. CAD tools test $d$ against $R \pm r$ to decide whether two arcs meet cleanly, which is the same check as Example 6.
The gear case is a clean real-world anchor: a gear train transmits motion only because adjacent pitch circles touch externally, a relationship formalised in the engineering standard for gear geometry. Get the centre-distance wrong and the whole train fails.
Mistakes to Watch For
Mistake 1: Checking only one of the two conditions
Where it slips in: A problem gives a small centre-distance and you test only $d = R + r$.
Don't do this: Conclude "no touch" the moment $d \neq R + r$.
The correct way: Compare $d$ against both $R + r$ and $|R - r|$. A distance that fails the external test often passes the internal one. The rusher who checks one formula and moves on misses every internal-tangency problem.
Mistake 2: Forgetting the absolute value in the internal condition
Where it slips in: Subtracting the radii in the wrong order.
Don't do this: Write $d = r - R$ and get a negative distance.
The correct way: Internal touch is $d = |R - r|$, the positive difference. Distance is never negative, so always take the larger radius minus the smaller.
Mistake 3: Assuming touching circles meet at two points
Where it slips in: Confusing "touching" with "intersecting."
Don't do this: Look for two common points on tangent circles.
The correct way: Touching means exactly one common point. Two common points is the intersecting case, where $|R - r| < d < R + r$. The second-guesser who redraws the figure hunting for a second crossing is chasing a point that is not there.
Conclusion
Circles touching each other meet at exactly one point, and the case is decided entirely by the centre-distance $d$.
External touch happens when $d = R + r$; internal touch happens when $d = |R - r|$.
The two centres and the point of contact are always collinear, and both circles share one tangent line there.
The most common mistake is testing only one condition; always compare $d$ against both $R + r$ and $|R - r|$.
Externally touching circles carry three common tangents, internally touching circles just one.
To work through tangency problems with a teacher, explore Bhanzu's geometry tutor, our high school math tutor sessions, or math classes online. To watch a trainer classify touching circles step by step, you can book a free demo class.
Read More
Tangents and normals — how tangent and normal lines behave around a circle.
Circles centered at the origin — the simplest coordinate circle, $x^2 + y^2 = r^2$.
Circumference of a circle — measuring the boundary the two circles meet along.
Parts of a circle — centre, radius, chord, and the other elements a tangency argument uses.
Center of a circle — the point whose separation decides how two circles meet.
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