AMC 8 2025 Problem 18
AMC8 2025 年第 18 题
圆★★★☆☆
The circle shown below on the left has a radius of 1 unit. The region between the circle and the inscribed square is shaded. In the circle shown on the right, one quarter of the region between the circle and the inscribed square is shaded. The shaded regions in the two circles have the same area. What is the radius , in units, of the circle on the right?
左下图的圆的半径为1个单位，阴影部分为圆和其内接正方形之间的区域。在右下图的圆中，阴影部分为圆和其内接正方形之间区域的四分之一。两个圆中的阴影区域面积相同。请问右下图的圆的半径R是多少个单位？
- A.
- B.2
- C.
- D.4
- E.
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
The area of the shaded region in the circle on the left is the area of the circle minus the area of the square, or . The shaded area in the circle on the right is of the area of the circle minus the area of the square, or , which can be factored as . Since the shaded areas are equal to each other, we have , which simplifies to . Taking the square root, we have The side length of the smaller square is , which follows from the Pythagorean Theorem since the square is inscribed in a circle of radius 1, making the diagonal equal to 2. Thus, the area of the square is . The circle in which the square is inscribed has radius 1, so its area is . Therefore, the area of the shaded region (the part of the circle not covered by the square) is . Now, consider the larger square, which is inscribed in a circle of radius . This square can be divided into four congruent right triangles by drawing lines from the center of the circle to the vertices of the square. Each triangle has legs of length and includes a angle at the center. Using the triangle area formula with a = b = R and , each triangle has area . Since there are four such triangles, the area of the square is . The area of the circle is , so the area of the shaded region (the part of the circle outside the square) is . We are told that this shaded region also has area equal to , so we set up the equation . Assuming , we divide both sides by , giving , which leads to , and finally R = 2. Thus, the answer is . ~ GREATEST ~
We start with the first area. Since the square is inscribed, its diagonal is its side length is its area is 2, therefore the first area is . The second area is , found in a similar manner. Writing and solving the equation, we have The answer is Since the two figures are similar, and the 4 regions in the first circle are equal in area to one of the four regions in the second circle, the second figure's area is 4 times the first figure's area. Therefore, the side lengths are multiplied by a factor of , so the answer is .
(Really similar to, but has a more larger explanation for the answer) First find the area of the smaller circle with the known radius. Since the radius is 1, the area of the circle is , and finding the side length of the square would equal to . Squaring that gives us the area of the square which is 2, and the shaded area of the first one is equal to . The second area can be shortened to , since we only need th of the sections. We then make them equal, which makes this equation: . Simplifying this equation would be . We can factor out the which makes . Dividing out the would give us 4 on the left side. which leaves us to choice .
We are given two circles, each with an inscribed square. In the first circle, the radius is 1 unit, and the region between the circle and the inscribed square is shaded. In the second circle, one quarter of the region between the circle and the inscribed square is shaded. The shaded areas in both circles are the same. Our goal is to determine the radius of the second circle. Step 1: Set up a ratio for the areas of the shaded regions We know that the shaded region in the first circle (with radius 1) and the shaded region in the second circle (with radius ) are proportional in terms of the total area between the circle and the inscribed square. The shaded area in the second circle is one quarter of the total area between the circle and the square, while the shaded area in the first circle is the entire region. Step 2: Relate the areas to the radius Let's recall that the area between the circle and the inscribed square is proportional to the square of the radius of the circle because the square's side length scales with the radius of the circle. So, in the first circle with radius 1: and the shaded area is proportional to the entire area between the circle and the square, i.e., 1. In the second circle with radius : and the shaded area is one-quarter of this total area, so it is proportional to . Step 3: Set up the ratio of shaded areas We are told that the shaded areas in both circles are the same, so we set up the following equation:
Step 4: Solve for Multiply both sides of the equation by 4: Taking the square root of both sides: R = 2 Thus, the radius of the second circle is .
Notice that if you shaded the rest of the region between the square and the circle, then the area would be 4 times the area of the shaded region in the smaller circle. Since the radius changes by a factor of , , solving for gets you 2. Hence, the answer is We want to find a quick and easy formula that can be applied to this (not necessarily a formula, but a rule). We take the area of the circle and subtract it from the area of the square and you will find the area of the 4 outer shaded regions. This rule is basically . We plug in the radius from the first figure and compute the answer to be . Since this represents the area of ONE region in the second figure, we have to multiply our result by four which gets us or to make it easier, you can factor the four and make it . Using our rule, we can set both equations equal to each other. , , R=2. Both area formulas, and are quadratic, therefore multiplying radius/side length by 2 gives the area. The shaded region of the larger figure is 4 times the smaller, therefore the answer is 2.
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