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)

$\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 60 \qquad \textbf{(D)}\ 75 \qquad \textbf{(E)}\ 80<math>

==Solution 1== Each of the side triangles has base length <math>2$ (Error compiling LaTeX. Unknown error_msg) 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

Solution 2

There are total squares in the diagram and each square can have triangles. Thus, the total number of triangles in the diagram is triangles. There are shaded triangles in the diagram (you can just count this up), so the percentage of the shaded triangles is . ~Pi_in_da_box

Video Solution 1 by SpreadTheMathLove

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