I’m not actually a fan of formulas. Instead, I encourage students to think critically about how formulas are derived. That way, these students are able to have a stronger intuitive sense of the way the math behind the formula works.
For instance, say you have a 30-60-90 triangle. Many students falter because they always mix up the sides, especially the side that takes the radical sign. Is it a , or a ?
To think of the proportions intuitively, simply remember that, in a 30-60-90 triangle, the shortest side is always half the length of the longest side. Therefore, we have a ,and a . The middle side will have to be less than , so it will either be , or .
As to which one, remember that a 30-60-90 triangle is a right triangle. So, if we use the Pythagorean theorem (which you should definitely be able to execute quickly and accurately), then the shortest side squared (which is 1) subtracted from the longest side squared is equal to 3 (in this case, I just assumed x is equal to 1 so that the shortest side and longest side are 1 and 2, respectively). Using the Pythagorean theorem, we square this number, and get a .
If you’ve followed me this far, now you have a way for testing the sides of a 30-60-90, instead of relying on a formula (which can be stressful because you may always forget it…just don’t forget the Pythagorean theorem).
Relying on formulas can also give us formula blindness. That is, even though we’ve remembered a formula, we try to apply it to a problem even when the problem is asking for something different. The reason students often fall into this trap is because a question may use language that is similar to the language you’d expect to conform to the formula.
Let’s take the following problem:
One side of a right triangle is twice the length of another side.
|Column A||Column B|
|The degree measure of the smallest angle of triangle||30|
- The quantity in Column A is greater
- The quantity in Column B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given
Your first inclination is to think, oh, it’s a 30-60-90 triangle. After all, I just told you that in a 30-60-90 triangle, the longest side is always twice the shortest. However, the GRE is always trying to trap you, especially on Quantitative Comparison. To avoid becoming a victim of their trap, you must think through the problem, instead of blindly relying on your formula.
With this problem, the twist is that the leg that is twice as long as the shortest leg could also be the middle leg. In that case, we would have a triangle that is not a 30-60-90 triangle. So, the shortest angle would be something different from 30 degrees (it is actually less than 30 degrees). Because Quantitative Comparison does not require you to know the exact number of that angle (something that would be beyond the scope of the GRE), the answer is (D).
Had the only possible scenario been a 30-60-90 triangle, then the answer would be (C). By thinking through the problem, and not simply relying on the formula, we were able to find a different possible triangle that conforms to the information in the problem (one side of a right triangle…).
That is not to say one should be able to intuit every formula. Doing so would only make things more complex. And, for the GRE quantitative section, we want, whenever we can, to make things easier for ourselves.
So, below are a few formulas, mostly from geometry, that you should memorize, especially if you are looking to score in the 700s.
Area of an equilateral triangle:
(where s is a side of the equilateral triangle)
Volume of a sphere:
Combined Work Rate:
Once you’ve memorized these formulas, you should practice them on relevant problems so that applying the formulas becomes natural. You should also be aware when the formula doesn’t completely apply, such as in the case above. Or, when you can find a way outside of the formula to solve the problem, i.e. just because a problem deals with a sphere does not mean you will have to rely on the formula above.
Ultimately, the GRE is testing the way you think. And simply plugging in a bunch of values to a set formula doesn’t test thinking skills insomuch as it tests your ability to memorize a formula. And, trying to memorize a formula is often more difficult than knowing how that formula was derived.
In upcoming posts, I’m going to go through and deconstruct formulas that I feel are laborious. Instead, I am going to help you understand how a formula was derived, and then show you some shortcuts, so you are not spending extra time plugging and chugging through some cumbersome formula.