Difference between revisions of "2005 AMC 8 Problems/Problem 25"
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Latest revision as of 21:13, 25 October 2020
Problem
A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?
Solutions
Solution 1
Let the region within the circle and square be . In other words, it is the area inside the circle the square. Let be the radius. We know that the area of the circle minus is equal to the area of the square, minus .
We get:
So the answer is .
Solution 2
We realize that since the areas of the regions outside of the circle and the square are equal to each other, the area of the circle must be equal to the area of the square.
So the answer is .
See Also
2005 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
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 |
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