CDS Mathematics · Circles

Touching Circles

Circles that touch have their point of contact on the line of centres: the centres are r₁ + r₂ apart outside, r₁ − r₂ apart inside.

All Circles notes CDS Mathematics strategy

Why this matters

Ten PYQs, three of them HARD. Five give the sum of the areas and the distance between the centres, which is a sum-and-sum-of-squares system for the radii. Two of the HARD ones put circles in an angle, where each centre lies on the bisector.

Concept 1 of 2: Radii from contact and areas

Touching externally, the centres are the two radii apart; touching internally, one radius minus the other. A sum of areas gives the sum of squares, and the identity

(a + b)^2 = a^2 + b^2 + 2ab

does the rest.

Definition

External contact: $d = r_1 + r_2$ . Internal contact: $d = r_1 - r_2$ .
Sum of areas $k\pi$ : $r_1^2 + r_2^2 = k$ .
Then $2r_1r_2 = (r_1 + r_2)^2 - k$ and $(r_1 - r_2)^2 = k - 2r_1r_2$ .
Three circles centred at a triangle's vertices, each touching the other two: the radii add to the semi-perimeter $s$ , and the one at $A$ is $s - a$ .

Difference of the radii

(r_1 - r_2)^2 = 2(r_1^2 + r_2^2) - (r_1 + r_2)^2

Worked example

Two circles touch externally, their centres are

11

cm apart, and their areas add to

61\pi

^2

. Find the radii.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (II) 2022 — Elementary Mathematics · Q88Moderate

Example 1 · Circles · Touching Circles

Two circles touch externally. The sum of their areas is

89\pi

square cm and the distance between their centres is 13 cm. What is the difference in their radii ?

Internal contact uses the difference

When one circle touches the other from inside, the centres are

r_1 - r_2

apart. Using the sum gives radii that do not fit the areas.

Diameters, not radii

Some items ask for the difference of the DIAMETERS, which is twice the difference of the radii.

Drill 7 more on radii from contact and areas

Concept 2 of 2: Circles inside an angle

A circle touching both arms of an angle is equally far from them, so its centre lies on the bisector. Its radius is its distance along the bisector times the sine of the half-angle.

Definition

A circle touching both arms of angle $2\theta$ at vertex $A$ : centre on the bisector, $r = AO \sin\theta$ .
Two such circles touching each other: $\dfrac{r_2}{r_1} = \dfrac{1 - \sin\theta}{1 + \sin\theta}$ .
A right angle ( $\theta = 45^\circ$ ): the centre of a circle of radius $r$ is $r\sqrt2$ from the corner, and $\dfrac{r_2}{r_1} = 3 - 2\sqrt2$ .
A $60^\circ$ angle ( $\theta = 30^\circ$ ): $\dfrac{r_2}{r_1} = \dfrac13$ .

Two circles in an angle

\dfrac{r_2}{r_1} = \dfrac{1 - \sin\theta}{1 + \sin\theta}

Worked example

Two circles in a

60^\circ

angle touch both arms and each other. The larger has radius

12

cm. Find the smaller.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2020 · CDS (I) 2020 — Elementary Mathematics · Q73Hard

Example 2 · Circles · Touching Circles

A circle of diameter 8 cm is placed in such a manner that it touches two perpendicular lines. Then another smaller circle is placed in the gap such that it touches the lines and the circle. What is the diameter of the smaller circle ?

Sine of the HALF-angle

The bisector splits the angle, so

r = AO\sin\theta

uses half the angle between the lines. With the full angle, a right angle would give

r = AO

Drill 1 more on circles inside an angle

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)

Radii from contact and areas
Difference of the radii
$(r_1 - r_2)^2 = 2(r_1^2 + r_2^2) - (r_1 + r_2)^2$
Circles inside an angle
Two circles in an angle
$\dfrac{r_2}{r_1} = \dfrac{1 - \sin\theta}{1 + \sin\theta}$

Watch out for (3)

Internal contact uses the difference
→ Radii from contact and areas
Diameters, not radii
→ Radii from contact and areas
Sine of the HALF-angle
→ Circles inside an angle

Test yourself on Circles

15 past CDS questions from this chapter, timed at 18 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 10 questions