2025 AMC 8

Complete problem set with solutions and individual problem pages

Problem 1 Easy

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?

  • A.

    40

  • B.

    50

  • C.

    60

  • D.

    75

  • E.

    80

Answer:B

Solution 1

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.

 

Solution 2

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 4^2 \cdot 2 = 32 triangles. There are 16 shaded triangles in the diagram, so the area of the star is \dfrac{16}{32} = \frac{1}{2} = \frac{50}{100}, or \boxed{\textbf{(B)}~50} percent.

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