AMC8 2025

AMC8 2025 · Q1

AMC8 2025 · Q1. It mainly tests Fractions, Area & perimeter.

The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire $4\times4$ grid is covered by the star?

下图所示的八角星是一种流行的绗缝图案。八角星占整个 $4\times4$ 网格的百分之多少？

(A) \ 40 \ 40

(B) \ 50 \ 50

(D) \ 75 \ 75

(E) \ 80 \ 80

Answer

Correct choice: (B)

正确答案：（B）

Solution

Each of the unshaded triangles has base length $2$ and height $1$, so they all have area $\frac{2 \cdot 1}{2} = 1$. 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 $4^2 - 4 \cdot 1 - 4 \cdot 1 = 8$. The star is then $\frac{8}{16} = \frac{1}{2} = \frac{50}{100}$, or $\boxed{\textbf{(B)}~50}$ percent of the entire grid.

每个未着色的三角形底边长 $2$，高 $1$，因此面积均为 $\frac{2 \cdot 1}{2} = 1$。每个未着色的单位正方形面积为 $1$。着色区域的面积等于整个网格面积减去未着色区域面积，即 $4^2 - 4 \cdot 1 - 4 \cdot 1 = 8$。因此，八角星占整个网格的 $\frac{8}{16} = \frac{1}{2} = \frac{50}{100}$，即 $\boxed{\textbf{(B)}~50}$%。

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