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Lucas Fink

SAT Math Practice: Geometry Challenge

Below is a question that integrates some of the SAT’s favorite areas of geometry with some of the more difficult skills tested—specifically, it’s about dealing with 3D objects.

This is a classic example of how the SAT makes a question more difficult. By integrating more than one topic into the question, the test-makers make more possibilities for errors on your part. Remember to work through the question methodically, and be careful not to make simple errors that will lead you astray.

Before you answer it, look back at those common SAT shapes linked above. You’ll need ‘em.


The question


In the figure above, the top face of the right cylinder is a circle with the point Q as its center. Point N lies on the edge of the cylinder’s base. The distance between points N and Q is twice the length of the radius of circle Q. If the volume of the cylinder is {8}pi*sqrt{3}, then what is the height of the cylinder?

  1. 2
  2. 2sqrt{3}
  3. 4
  4. 4sqrt{3}
  5. 8

Before going any further, try working through the question alone. Draw the figure on a piece of paper so you can work with it as if this were your actual SAT. You’ll want to be able to draw on the figure to complete it, so just looking at your screen isn’t good enough.


Step one: Draw in the extra information

Draw lines that the figure doesn’t show. The question specifically mentions the distance between N and Q, so draw that segment in.


We also need to draw in a radius. You could draw it in any number of directions, but there’s one placement that will give us a familiar shape to work with when combined with segment NQ:


Remember to keep an eye out for right triangles.

Oh, and there’s one last thing to add to the figure from the question—the measurements.


Those line measurements may not be very concrete, but seeing them written into the figure like that might help you to see an important little detail…


Step two: Check your goal

What is the question asking for? Make sure you don’t lose sight of that and fall into any traps… We’re looking for the height of the cylinder, which is, conveniently, one of the legs of the right triangle we just drew. Noooo problem.

And hey, there’s a shortcut! Those two sides we wrote in look like sides of a 30˚-60˚-90˚ triangle, which the SAT provides at the beginning of each math section.

And since it’s a right triangle, we know that’s got to be true. That means that the height—the other leg of the triangle—must be r{sqrt{3}}.


Step three: Write out any equations you can

Also, from the beginning of each math section (and from your head, ideally), we know that the volume of the cylinder, V, must be {pi}{r^2}{h}. So let’s write those formulas out using the information about our figure we have.


h= r{sqrt{3}}

You don’t have to use the 30˚-60˚-90˚ triangle info, although it does make things faster. You could use the Pythagorean theorem for that second equation:

h^2= r^2+(2r)^2

But for brevity’s sake, let’s use the shortcut.


Step four: Cancel and substitute

Finally, let’s take those equations and make them simpler. Remember that we’re trying to isolate h, the height.


Take out pi from either side…


Plug in {r}sqrt{3} for h, as we saw from that triangle…


Cancel out the sqrt{3} from either side…


If {r^3}=8, then r=2


Step five: Double check your goal

Even though we found r, we’re not done. And notice, by the way, that answer choice (A) is 2, just in case you’re too hasty. Instead, make sure you look back and see that we want h, which is 2{sqrt{3}}. That’s (B). Now you’re done.


Remember to take it step by step

The SAT rewards caution, so be wise and take it one step at a time. Even higher level problems like this one start to fall into place if you just work with the immediate information, bit by bit.


About Lucas Fink

Lucas is the teacher behind Magoosh TOEFL. He’s been teaching TOEFL preparation and more general English since 2009, and the SAT since 2008. Between his time at Bard College and teaching abroad, he has studied Japanese, Czech, and Korean. None of them come in handy, nowadays.

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