Are you solid on angles?

Geometry Level 4

At the home of a mathematician friend I noticed an unusual holiday ornament the other day that consists of a sphere and a concentric cube. It would have been inappropriate to take a picture under the circumstances, but, fortunately, I found a somewhat similar picture online (the ornament comes in more pleasing colours, neon pink and purple, though). If exactly three quarters of the surface area of the sphere are exposed, find the (non-oriented) solid angle $\Omega$ subtended by the exposed part of the cube's surface at the center of the cube.

As your answer, submit the integer nearest to $1000\Omega$ .

Suggested by Cde Huan Bui

The answer is 3142.

1 solution

Steven Chase
Dec 28, 2018

The solid angle for an entire sphere is $4 \pi$ . Since the sphere is three-quarters exposed, it gets $3 \pi$ of that, leaving $\pi$ for the exposed cube surface area. Multiplying $\pi$ by $1000$ and rounding to the nearest integer yields $3142$

Yes, exactly! Thank you! It's trivial; I just want to see whether people understand the notion of a solid angle (it's more of a physics than a maths concept).

Otto Bretscher - 2 years, 5 months ago

Thanks. By the way, I liked your final ornament problem. Regrettably, I got it wrong because I had the wrong picture in my head (with the sphere small enough to use a spherical cap).

Steven Chase - 2 years, 5 months ago

That one was really inspired by your problems on the intersection of spheres and cubes.

Otto Bretscher - 2 years, 5 months ago

Are you solid on angles?

The answer is 3142.

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