2025 AMC 8 Problems/Problem 12
Contents
[hide]Problem
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?
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 points. By the Pythagorean Theorem, the distance from the center to one of these
points is
, so the area of this circle is
.
~Soupboy0
Solution 2 (very unrecommended)
Draw the circle in the grid and analyze the radius. It's radius is a little more than 2 but a lot less than 2.5, so the area is a little more than 4π. So, the area of the circle is with a radius of approximately 2.23.
-- leafy
Video Solution (A Clever Explanation You’ll Get Instantly)
https://youtu.be/VP7g-s8akMY?si=Oe8Ka0kEZiLNXA5g&t=1096 ~hsnacademy
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=jTTcscvcQmI
Video Solution 2 by Thinking Feet
See Also
2025 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.