2021 WSMO Accuracy Round Problems/Problem 5
Problem
Suppose regular octagon has side length
If the distance from the center of the octagon to one of the sides can be expressed as
where
and
is not divisible by the square of any prime, find
Solution 1
[asy]
size(150);
draw(polygon(8)); for(int i=45; i<=360; i+=45){ dot(rotate(22.5)*dir(i)); }
label("",dir(337.5),SE);
label("
",dir(22.5),NE);
label("
",dir(67.5),N);
label("
",dir(292.5),S);
label("
",dir(247.5),S);
label("
",dir(202.5),W);
label("
",dir(157.5),NW);
label("
",dir(112.5),N);
label("
",(0,0),S);
dot((0,0),red);
draw(dir(112.5)--dir(67.5)--(0,0)--cycle,red);
dot(dir(112.5),red);
dot(dir(67.5),red);
label("
",(0,1.05));
draw(anglemark(dir(67.5),(0,0),dir(112.5)),red);
label("
",(0,0.25),N);
[/asy]
Let the center of the octagon be
We will focus on triangle
Let
From the Law of Cosines on triangle
we find that
Now, let the distance from the center of the octagon to one of its sides be
This means that
In addition, from the sine area formula,
Therefore, we have
~pinkpig
Solution 2
Note that the area of a polygon with sides,
side length, and
apothem (distance from the center to one of the sides) can be expressed as
Applying this formula, we get
Now, we need something to equate to this. Remember that the area of a regular octagon with side length
is
This means that the area of octagon
is
Therefore, the answer is
~captainnobody