2025 AMC 8 Problems/Problem 1

Problem

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?

path x = (0,1)--(1,2)--(2,2)--(1,1)--cycle;
path y = reflect((0,0),(4,4)) * x;

fill(x, gray(0.6));
fill(rotate(90, (2,2)) * x, gray(0.6));
fill(rotate(180, (2,2)) * x, gray(0.6));
fill(rotate(270, (2,2)) * x, gray(0.6));

fill(y, gray(0.8));
fill(rotate(90, (2,2)) * y, gray(0.8));
fill(rotate(180, (2,2)) * y, gray(0.8));
fill(rotate(270, (2,2)) * y, gray(0.8));

draw((1,1)--(q293579105798023
));
draw((3,1)--(1,3));

add(grid(4,4));

path w = (1,0)--(2,1)--(3,0);

draw(w);
draw(rotate(90, (2,2)) * w);
draw(rotate(180, (2,2)) * w);
draw(rotate(270, (2,2)) * w);
 (Error making remote request. Unknown error_msg)

Solution 1

Each of the side triangles has base length and height , so they all have area . Each of the four corner squares has side length and hence area . Then the area of the shaded region is . Since , our answer is . ~cxsmi

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=jTTcscvcQmI