Theorem

Distance between the centres of touching circles

Geometry • Circle • Touching Circles

Formula

d = r_{1} + r_{2} (external), d = ∣ r_{1} - r_{2} ∣ (internal)

Two circles of radii

r_{1}

and

r_{2}

that touch externally have their centres

r_{1} + r_{2}

apart, and two that touch internally have them

∣ r_{1} - r_{2} ∣

apart; in both cases the point of contact lies on the line of centres.

How to use it

1Read 'touch externally' as the single equation $r_{1} + r_{2} = d$ , which is usually one of the two conditions a question supplies.
2Pair it with a second condition on the radii, such as their squares or their product, and solve as a sum-and-product system.
3Circles that intersect at two points have $∣ r_{1} - r_{2} ∣ < d < r_{1} + r_{2}$ , which bounds the configuration.

Worked example

Two circles of radii

4

cm and

9

cm touch externally. How far apart are their centres?

Step-by-step

1
$d = r_{1} + r_{2} = 4 + 9$
Why: For external contact the point of contact lies between the centres.
2
$= 13$ cm

Answer

13

Watch out

External and internal contact give different relations, and a question that says only 'touch' leaves both open. Halving the centre distance gives a radius only when the two circles are equal.

Also known as

circles touching externally, distance between centres, internally touching circles, line of centres

Appears in past papers

Formula

Worked example

Appears in past papers

2020 (II)

2020 (I)

2018 (II)

2017 (II)

2017 (I)

2016 (II)