Find the area swept out by an unit square when rotated 45 degrees about a vertex.
A square the union of four line segments. The area swept out can be broken down into two half-squares plus a 45 degree sector minus a kite.
The two half-squares have the area of the original square = 1
The sector is pi*sqrt(2)^2/8 = pi/4
The kite is a right kite with side lengths 1 and sqrt(2)-1. Area = sqrt(2)-1
The final area is then 1 + pi/4 - (sqrt(2)-1) = 2 + pi/4 - sqrt2
For fun, I whipped up a demonstration:
https://www.desmos.com/calculator/m2t4ayw9md
(It by 'square' you mean to include the interior, then don't subtract the kite.)
|
|
Posted by Jer
on 2023-11-23 09:28:39 |
No Subject