Walking In Triangles
Brilli the Ant woke up one day on a mysterious, smooth, spherical planet. Lost, Brilli began to walk in a straight line. Brilli walked 12 miles before turning 90 degrees left. After a while of walking in a straight line, Brilli lost track of how far he had gone, so he turned 90 degrees left again. After another 12 mile walk, Brilli found himself right back where he started! (He had not crossed his own path at any point before this.)
What is the surface area of the planet Brilli is on, in square miles?
Give your answer to 3 decimal places.
The answer is 733.3859.
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In spherical geometry, if a triangle has two right angles, the two distances opposite those angles are quadrants. Thus, 12 miles is one-fourth the circumference of the spherical planet.
Thus, the total circumference of the planet is 48 miles, and its radius is π 2 4 .
Its surface area is 4 π r 2 = 4 π ( π 2 4 ) 2 ≈ 7 3 3 . 3 8 5 9