Problem:
A unit square is rotated 45β about its center. What is the area of the region swept out by the interior of the square?
Answer Choices:
A. 1β22ββ+4Οβ
B. 21β+4Οβ
C. 2β2β+4Οβ
D. 22ββ+4Οβ
E. 1+42ββ+8Οβ
Solution:
Let O be the center of unit square ABCD, let A and B be rotated to points Aβ² and Bβ², and let OAβ² and Aβ²Bβ² intersect AB at E and F, respectively. Then one quarter of the region swept out by the interior of the square consists of the 45β sector AOAβ² with radius 22ββ, isosceles right triangle OEB with leg length 21β, and isosceles right triangle Aβ²EF with leg length 22ββ1β. Thus the area of the region is
