SSC CGL Maths · Circles

Touching Circles

When two circles touch, the distance between their centres is the sum or the difference of the radii, and for circles touching externally the direct common tangent shrinks to 2√(r₁r₂).

All Circles notes

Why this matters

Twelve questions in the bank, 9 EASY and 3 MODERATE. Nine of them ask for the direct common tangent of two circles that touch externally, which is one square root once you know the shortcut. The other three use the distance between the centres, or the right angle that the point of contact makes with the ends of a direct tangent.

Concept 1 of 2: Direct common tangent of two circles touching externally

Circles that touch externally have their centres exactly

r_1 + r_2

apart. Put that distance into the direct tangent formula and the two squares collapse to a single product.

Definition

Touching externally means $d = r_1 + r_2$ .
The direct tangent then has $L^2 = (r_1 + r_2)^2 - (r_1 - r_2)^2 = 4r_1 r_2$ , so $L = 2\sqrt{r_1 r_2}$ .
The shortcut is safe only when the circles touch externally. For circles apart or cutting each other, go back to $\sqrt{d^2 - (r_1 - r_2)^2}$ .
Run backwards: $r_1 r_2 = \frac{L^2}{4}$ , so one radius and the tangent give the other radius.
A question that says only "their common tangent" for circles touching externally means a direct one: the third tangent, at the point of contact, has no length between two points of contact.

Touching circles, direct tangent

L = 2\sqrt{r_1 r_2}

Lthe direct common tangent between its two points of contact
$r_1, r_2$ the radii of the two touching circles

Worked example

Two circles of radii 9 cm and 4 cm touch each other externally. Find the length of their direct common tangent.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 2 · Q70Moderate

Example 1 · Circles · Touching Circles

A circle with radius

x

touches another circle with radius

2x

externally. What is the length of a direct common tangent?

Using $2\sqrt{r_1 r_2}$ for circles that do not touch

The shortcut assumes the centres are exactly

r_1 + r_2

apart. When the distance is given and is different, use

\sqrt{d^2 - (r_1 - r_2)^2}

Taking the sum of the radii as the tangent

r_1 + r_2

is the distance between the centres of touching circles. The tangent is

2\sqrt{r_1 r_2}

, which is never longer than that.

Drill 8 more on direct common tangent of two circles touching externally

Concept 2 of 2: Centres, the point of contact, and a circle touching two others

Two touching circles share a tangent at the point of contact, and both radii to that point are perpendicular to it. So the two centres and the point of contact lie on one straight line, and the distance between the centres is found by adding or subtracting radii.

Definition

Touching externally: $d = r_1 + r_2$ . Touching internally: $d = r_1 - r_2$ . The point of contact lies on the line of centres.
The least distance between centres for exactly three common tangents is $r_1 + r_2$ : any closer and the circles cut, with only two.
For circles touching externally at P, with a direct common tangent touching them at A and B: the tangent at P meets AB at M, and the tangents from M are equal, so $MA = MP = MB$ . Hence $\angle APB = 90^\circ$ .
A third circle of radius $r$ touching two circles externally has its centre $r_1 + r$ and $r_2 + r$ from their centres. Put those lengths into the triangle the question describes.

Distance between touching centres

d = r_1 + r_2 \text{ (external)} \qquad d = r_1 - r_2 \text{ (internal)}

dthe distance between the centres

Worked example

Circles with centres A and B have radii 9 cm and 2 cm, and AB = 13 cm. A third circle with centre C and radius r cm touches both externally, and

\angle ACB = 90^\circ

. Find r.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 13 Sep 2024, Shift 1 · Q63Moderate

Example 2 · Circles · Touching Circles

A and B are centres of two circles with radii 2 cm and 1 cm respectively, where AB = 5 cm. C is the centre of another circle of radius r cm, which touches each of the above two circles externally. If

\angle ACB = 90^\circ

, then the value of r is:

Adding the radii for circles touching internally

Circles touching internally have one inside the other, so the distance between their centres is the difference of the radii.

Using a radius alone for the distance to a third touching circle

A circle of radius

r

touching a circle of radius

r_1

externally has its centre

r_1 + r

away, the sum of both radii.

Drill 2 more on centres, the point of contact, and a circle touching two others

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Direct common tangent of two circles touching externally
Touching circles, direct tangent
$L = 2\sqrt{r_1 r_2}$
Centres, the point of contact, and a circle touching two others
Distance between touching centres
$d = r_1 + r_2 \text{ (external)} \qquad d = r_1 - r_2 \text{ (internal)}$

Watch out for (4)

Using $2\sqrt{r_1 r_2}$ for circles that do not touch
→ Direct common tangent of two circles touching externally
Taking the sum of the radii as the tangent
→ Direct common tangent of two circles touching externally
Adding the radii for circles touching internally
→ Centres, the point of contact, and a circle touching two others
Using a radius alone for the distance to a third touching circle
→ Centres, the point of contact, and a circle touching two others

Test yourself on Circles

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Circles chapter test Drill all 12 questions