2025 AMC 8

Complete problem set with solutions and individual problem pages

Problem 12 Medium

The region shown below consists of 24 squares, each with side length 1 centimeter. What is the area, in square centimeters, of the largest circle that can fit inside the region, possibly touching the boundaries?

  • A.

    3\pi

  • B.

    4\pi

  • C.

    5\pi

  • D.

    6\pi

  • E.

    8\pi

Answer:C

Solution 1

The largest circle that can fit inside the figure has its center in the middle of the figure and will be tangent to the figure in 8 points. By the Pythagorean Theorem, the distance from the center to one of these 8 points is \sqrt{2^2 + 1^2} = \sqrt5, so the area of this circle is \pi \sqrt{5}^2 = \boxed{\textbf{(C) } 5\pi}.

 

Solution 2

Draw the circle in the grid and analyze the radius. Its radius is a little more than 2 but a lot less than 2.5, so the area is a little more than 4\pi . So, the area of the circle is \boxed{\textbf{(C) } 5\pi} with a radius of approximately 2.23.

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