Problem:
A unit square is rotated 45∘ about its center. What is the area of the region swept out by the interior of the square?
Answer Choices:
A. 1−22+4π
B. 21+4π
C. 2−2+4π
D. 22+4π
E. 1+42+8π
Solution:
Let O be the center of unit square ABCD, let A and B be rotated to points A′ and B′, and let OA′ and A′B′ intersect AB at E and F, respectively. Then one quarter of the region swept out by the interior of the square consists of the 45∘ sector AOA′ with radius 22, isosceles right triangle OEB with leg length 21, and isosceles right triangle A′EF with leg length 22−1. Thus the area of the region is
