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So this came up tonight with a friend while we were studying, he had an old SAT question that asked "How many sides does this shape have?". Below is a representation of the image provided.

After searching around and debating it, we came up with nothing. A polygon is defined as "consisting of a number of points and an equal number of line segments, with no three successive points collinear". But the question clearly stated "shape" and not polygon. Because of this, we're pretty stumped.

We then concluded on the proposition that this would have to depend on whether or not you were taught if lines have 1 or 2 sides.

a pentagon lollipop

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  • $\begingroup$ I tried to put the graphic in but was told we don't support this format. It is a regular pentagon with a line segment attached to one vertex. $\endgroup$ – Ross Millikan Oct 16 '17 at 4:52
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    $\begingroup$ "Side" is ambiguous. I'd say it has two sides – the inside, and the outside – but that's probably not what they want. $\endgroup$ – Gerry Myerson Oct 16 '17 at 4:58
  • $\begingroup$ @GerryMyerson Thanks for that... your comment was the highlight of my day :) $\endgroup$ – user7530 Oct 16 '17 at 5:00
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This is a terrible question unless it comes with definitions of "shape" and "sides". If you have those, you should be able to find an unambiguous answer. As there are not (as far as I know) standard mathematical definitions of those terms, there is no excuse for the SAT asking the question. I would define a side as a line segment that divides the inside from the outside and get $5$. You imply arguments for $6$ and $7$, both of which are reasonable answers. The fact that there are at least two reasonable answers proves it is not a good question.

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This is a horrible question and it has no business being on a test used for college admissions (or any other "serious" purpose.)

You could easily make the case for 6 or 7 sides (and probably with a bit more effort could argue for some other number of sides too).

The case for 6: if you call a "shape" a polyhedral complex and a "side" a one-dimensional face, there are six sides in the complex illustrated in the figure.

The case for 7: if you interpret "shape" as polygon, the simplest (but not only) interpretation of the figure as a polygon is that it is heptagon with two coincident vertices.

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