Question

The diagram given alongside represents three circular garbage cans each of unit radius. The three cans are touching each other as shown. Find, in metres, the perimeter of the rope encompassing the three cans and hence the area of the circumscribing circle.

Comments

How is AB be 2 cm if the angle is 60 in between the aopb also it is not a equal side

---

Because, the rope is worked as a tangent to circles, therefore, angle BAO and ABP are right angles and quadrilateral OABP forms a rectangle. Since OP = 2 cm. Thus, AB = 2 cm.

Answer 1

AB = CD = EF = Distance between two radii = 2 cm

∠AOF = ∠BPC = ∠DQE = 120º

∴ Perimeter of the circumscribing figure 

= AB + CD + EF + Circumferences of sectors (FOA + BPC + DQE) 

Since three equal sectors of 120º = 1 full circle of the same radius.

∴ Perimeter of the circumscribing figure = 2πr + 2 + 2 + 2 

= (2πr + 6) cm 

= (2π + 6) cm 

If R is the radius of the circumscribing circle, then, 

2πR = 2π + 6 = (2 × 3.14 + 6) cm = 12.28 cm

⇒ R = \(\frac{12.28}{2 \times 3.14}\) = 2 cm.

∴ Area of circumscribing circle = πR² = (3.14 × 4) cm² 

= 12.56 cm².