Problem:
Haruki has a piece of wire that is centimeters long. He wants to bend it to form each of the following shapes, one at a time.
Which of the shapes can Haruki make?
Answer Choices:
A. Triangle only
B. Hexagon and square only
C. Hexagon and triangle only
D. Square and triangle only
E. Hexagon, triangle, and square
Solution:
To make a regular hexagon with side length , Haruki would need of wire, which is more than he has.
To make a square of area , the side length of the square would be . The perimeter of a square with side length is , so Haruki has enough wire to make the square.
A right triangle with two legs of and long, would have a hypotenuse of long (a triangle is a triangle with all the side lengths multiplied by ). Thus, the total perimeter of the triangle is , so Haruki also has enough wire to make the triangle.
Therefore, the answer is .
The problems on this page are the property of the MAA's American Mathematics Competitions