AMC 8 2025 Problem 1
AMC8 2025 年第 1 题
面积与周长（综合）★☆☆☆☆
The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire grid is covered by the star?
下图所示的八角星是一种流行的拼布图案。请问这个星形图案的面积占整个4×4网格的百分之多少？
- A.40
- B.50
- C.60
- D.75
- E.80
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
we count the amount of shaded half triangles, 8. we half this to get the amount of full squares it equals. then we add 4 for the four full squares in the middle. this gets us a total of eight full squares, out of a total of 16. so 8 / 16 is simplified to 1 / 2 or percent of the entire grid. Each of the unshaded triangles has base length 2 and height 1, so they all have area . Each of the unshaded unit squares has area 1. The area of the shaded region is equal to the area of the entire grid minus the area of the unshaded region, or . The star is then = = , or percent of the entire grid. —-DummyAMC
There are 16 total squares in the diagram and each square has 2 triangles whose areas are half the area of a unit square. Thus, the total number of triangles in the diagram is triangles. There are 16 shaded triangles in the diagram, so the area of the star is , or percent.
There are 4 squares that are entirely shaded and 4 squares that have no shading. This cancels them out. The rest of the squares are half-half. Therefore the shaded region is percent of the grid.
Note that we can move one triangle from each of the four cells in the middle to each of the four corners. This will leave every cell in the grid with one triangle each, and each triangle has an area of half the area of each cell. Thus, our answer must equal to , and so our answer is.
The shaded area is a 2×2 square in the middle of the figure combined with 8 small triangles. Since each small triangle is of a unit square, the star's area is equal to the area of unit squares.
You can move the extra half a squares and make 4 squares so there are . The answer is percent of 16. William
We can use the Pick's Theorem: . Plug that in and you get . The 2 1's cancel, so you get . Since it's a 4 x 4 grid, the total area is 16. = , and percent. Thus the answer is.
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