Question

One medium circle and one small circle touch each other, and each circle touches the larger circle.

The figure shows two circles of different radius inscribed in a larger circle. The two circles are drawn such that both touch the circle as well as each other without overlapping. The radius of the larger circle inscribed in the circle is labeled as 9 centimeters and the radius of the smaller circle inscribed in the circle is labeled as 4 centimeters. The region inside the circle not covered by the two circles is shaded.

Which is the area of the shaded region?

A.
72
π
square centimeters

B.
97
π
square centimeters

C.
26
π
square centimeters

D.
169
π
square centimeters

Answer 1

The area of shaded is 72π cm² ⇒ answer A

Explanation:

* Lets explain how to solve the problem

- The area of any circle is A = πr², where r is the radius of the circle

- The length of the radius of a circle is 1/2 the length of its diameter

- From the figure:

∵ There are two inscribed circles touch the larger circle internally

and touch each other externally

∴ The centers of the three circles are collinear

∴ The diameter of the larger circle = the sum of the diameters of

the two inscribed circles

∵ The radii of the inscribed circles are 4 cm and 9 cm

∵ The diameter = twice the radius

∴ The diameters of the inscribed circles are 2 × 4 = 8 cm and

2 × 9 = 18 cm

∴ The diameter of the larger circle = 8 + 18 = 26 cm

∵ The radius of the circle = 1/2 diameter

∴ The radius of the larger circle = 1/2 × 26 = 13 cm

- The area of the shaded part is the difference between the

area of the larger circle and the sum of the areas of the two

inscribed circles

∵ A = πr²

∴ The area of the larger circle = π(13)² = 169π

∵ The sum of areas of the inscribed circles = π(4)² + π(9)²

∴ The sum of areas of 2 circles =16π + 81π = 97π

∵ Area of shaded = A larger circle - sum of A of 2 circles

∴ Area of shaded = 169π - 97π = 72π

* The area of shaded is 72π cm²