Three sectors of a circle are removed from a regular hexagon to form the shaded shape shown.
Each sector has perimeter 18 mm. What is the perimeter, in mm, of the shaded shape formed?
A. 48 B. 50 C. 52 D. 54 E. 56
We let s be the length, in mm, of the sides of the hexagon.Each of the three sectors has two straight edges of length s.
Because each of these sectors has the same perimeter, it follows that the arcs that are parts of the perimeters of these sectors all have the same length. We let c be this length, in mm.
The perimeter of each sector is 18mm. Therefore 2s + c = 18.
We see from the diagram that the perimeter of the shaded shape is made up of six straight edges and three arcs.
It follows that the perimeter of the shaded shape, in mm, is given by
6s + 3c = 3(2s + c) = 3 × 18 = 54.
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