Circular reasoning

Geometry Level 3

Given points A A and B B in three space, such that ∣ A B ‾ ∣ = 6 \left |\overline{AB} \right | = 6 , what is the area traced out by all possible choices of point C C such that A C ‾ ⊥ B C ‾ \overline{AC} \perp \overline{BC} ?

Provide your answer to the nearest integer.

If you think that the locus of all possible points C C is a volume, put -1 as your answer, and if you think the area is infinite, put -2 as your answer.


The answer is 113.

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Aug 2, 2017

C C must lie on the surface of a sphere whose diameter is A B AB .

Therefore, the area traced out by all possible values of C C is the surface area of a sphere with radius 3.

⟹ A = 4 π r 2 = 36 π ≈ 113 \implies A = 4\pi r^2 = 36\pi \approx \boxed{113}

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