In 3-dimensional space, what is the set of all points 12 units from the origin?

Question

I have come across this question in an ACT practice paper, Form 0057B. The only search result I can find is an explanation on a website called crackacc, but it has become a broken link. The options are: a circle, a sphere, a line, a cylinder, and 2 parallel lines. My answer was a cylinder, but I got wrong and I have been trying to visualize what the resulting shape would be but I cannot come to a solution.

Answer 1

It is a sphere. If you know that in two dimensions that it would have been a circle, then you can reason as follows. Imagine looking at the object from the side, or from above, and what would you see? A circle. From any direction you would see a circle. And the things that are circles from every direction are spheres.

Answer 2

Aaron's answer is good but I wish to add two elaborations (I don't have much history here so I can't leave a comment).

First, let's speak directly to "I have been trying to visualize what the resulting shape would be". Preamble: Bang a nail into a piece of paper on a wood board. Tie one end of a 12 cm string to nail, other end to a pen. Drawing on the paper with the string taut you get a familiar shape, right? Now, instead of a nail on a board, tie the string to the top of a flagpole; keeping the string taut, try to make the free end visit every allowed position, "drawing" invisibly in the air; what shape is that?

Second, the allowed answers in the test question bother me. Sphere almost always refers to a solid 3-d object, which includes (infinitely) many points closer to the origin. The correct name of the shape we are discussing is spherical shell. (You may have wrongly remembered or reported the original wording; it's also possible that the people who make these tests are way less mathematically capable than those of fifty years ago.)