Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown. [asy]

unitsize(1.5 cm);

pair[] A;

for(int i = 1; i <= 10; ++i) {

A[i] = dir(90 - 360/10*i);

draw(dir(90 - 360/10*i)--dir(90 - 360/10*(i + 1)));

}

draw(A[10]--(A[10] + A[2] - A[3])--(A[10] + A[2] - A[3] + A[4] - A[5])--(A[10] + A[2] - A[3] + A[4] - A[5] + A[2] - A[1])--(A[10] + A[2] - A[3] + A[4] - A[5] + A[2] - A[1] + A[4] - A[3]));

draw((A[10] + A[2] - A[3])--(A[10] + A[2] - A[3] + A[10] - A[1])--(A[10] + A[2] - A[3] + A[10] - A[1] + A[8] - A[9])--(A[10] + A[2] - A[3] + A[10] - A[1] + A[8] - A[9] + A[6] - A[7]));

dot(A[10]);

[/asy]

A square, a regular pentagon, and a regular $n$-gon, all with the same side length, also completely surround a point. Find $n$.