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CDS Mathematics · Mensuration 2D

Touching Circles

Circles that touch each other or the sides of a figure: join the centres, and the distance between them is the sum or the difference of the radii.

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Why this matters

A small page, but the hardest in the chapter: most of its questions are HARD. They all yield to the same first move, so the page is short and the payoff is large.

Concept 1 of 2: Join the centres

The point where two circles touch lies on the line joining their centres. So the distance between the centres is the sum of the radii (touching from outside) or their difference (one inside the other). Turn the figure into a triangle of centres and use Pythagoras.

Definition

  • Touching externally: O1O2=r1+r2O_1O_2 = r_1 + r_2. Touching internally: O1O2=R−rO_1O_2 = R - r.
  • A circle touching a straight side has its centre at a distance rr from that side.
  • Three equal circles of radius rr touching each other: centres form an equilateral triangle of side 2r2r; the circle round all three has radius r+2r3r + \dfrac{2r}{\sqrt3}.
  • Four equal circles of diameter DD touching in a square: the gap in the middle holds a circle of diameter D(2−1)D(\sqrt2 - 1).
  • Two equal circles cut from a disc of radius RR are largest at radius R2\dfrac R2, leaving half the disc.

Distance between centres

O1O2=r1+r2  (outside),O1O2=R−r  (inside)O_1O_2 = r_1 + r_2 \;(\text{outside}), \qquad O_1O_2 = R - r \;(\text{inside})

Worked example

Two circles in a 1010 cm by 88 cm rectangle: the larger touches the top, bottom and left sides, and the smaller touches the bottom side, the right side and the larger circle. Find the smaller radius.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2024 · CDS (II) 2024 — Elementary Mathematics · Q84Hard

Example 1 · Mensuration 2D · Touching Circles

The area of the circle circumscribing three identical circles touching each other is π(2+3)23\frac{\pi\left(2 + \sqrt{3}\right)^2}{3} square cm. What is the radius of one of the smaller circles ?

Reject the root that doesn't fit the figure

The touching condition is a quadratic, and one root is usually impossible: a circle wider than the rectangle, or one that would overlap the other. Check the root against the figure before choosing.
Drill 6 more on join the centres

Concept 2 of 2: The gap between touching circles

The curved region enclosed by circles that touch is the polygon of centres minus the sectors of the circles inside it. For three equal circles the three sectors are 60∘60^\circ each, which together make half a circle.

Definition

  • Three equal circles of radius rr: gap =3r2−πr22=r22(23−π)= \sqrt3 r^2 - \dfrac{\pi r^2}{2} = \dfrac{r^2}{2}(2\sqrt3 - \pi).
  • Four equal coins at the corners of a square of side 2r2r: the four quarter-circles make one whole circle, so the uncovered part is 4r2−πr24r^2 - \pi r^2.
  • Circles of different radii at the corners of a triangle: sector at each corner =angle360∘πr2= \dfrac{\text{angle}}{360^\circ}\pi r^2; the three angles add to 180∘180^\circ.
  • Radii of three mutually touching circles centred at the vertices of a triangle with sides a,b,ca, b, c: s−as - a, s−bs - b, s−cs - c.

Three equal touching circles

gap=34(2r)2−3⋅60∘360∘πr2=r22(23−π)\text{gap} = \frac{\sqrt3}{4}(2r)^2 - 3\cdot\frac{60^\circ}{360^\circ}\pi r^2 = \frac{r^2}{2}(2\sqrt3 - \pi)
2r60°60°60°

Gap = triangle of centres − three 60° sectors = √3 r² − ½ π r².

Worked example

Three circles of radius 77 cm touch one another. Find the area enclosed between them. (π=227, 3=1.732)(\pi = \tfrac{22}{7},\ \sqrt3 = 1.732)
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (I) 2022 — Elementary Mathematics · Q18Hard

Example 2 · Mensuration 2D · Touching Circles

What is the area of the region enclosed by three identical circles (each of radius 4 cm) touching each other?

The sectors add to half a circle, not a whole one

Three 60∘60^\circ sectors make 180∘180^\circ, half a circle. Subtracting a whole circle gives a negative gap, and an option built on it is usually printed.
Drill 5 more on the gap between touching circles

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)

  • Join the centres

    Distance between centres

    O1O2=r1+r2  (outside),O1O2=R−r  (inside)O_1O_2 = r_1 + r_2 \;(\text{outside}), \qquad O_1O_2 = R - r \;(\text{inside})
  • The gap between touching circles

    Three equal touching circles

    gap=34(2r)2−3⋅60∘360∘πr2=r22(23−π)\text{gap} = \frac{\sqrt3}{4}(2r)^2 - 3\cdot\frac{60^\circ}{360^\circ}\pi r^2 = \frac{r^2}{2}(2\sqrt3 - \pi)

Watch out for (2)

Test yourself on Mensuration 2D

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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