What is k = ? k =?

Geometry Level 2

The above diagram has 6 6 circles of radius 4 4 , one circle of radius 8 8 , and one circle of radius 16 16 . The shaded area is k π k\pi . What is the value of k = ? k = ?

Figure not drawn to scale.

125 128 200 225 135

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Hana Wehbi
Aug 17, 2017

Area of the large gray circle: 1 6 2 π = 256 π 16^2\pi = 256 \pi .

Area of the small white 5 5 circles = 5 × ( 16 π ) = 80 π 5\times (16\pi)= 80\pi and area of one gray small circle is 16 π 16\pi .

Area of medium white circle = 8 2 π = 64 π ⟹ 8^2\pi= 64\pi \implies

Area of the gray shaded region = ( 256 − 80 − 64 + 16 ) π = 128 π ⟹ k = 128 (256 - 80 - 64+ 16)\pi = 128\pi \implies k=\boxed{128} .

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