There are other valid approaches here as well, but I think those two will be the ones that most people find.
This question presents me with another opportunity to remind you of what’s important to take away from these discussions. The goal of going over this solution is not to teach you a formal way to approach questions that ask about the border tiles on square game boards, because there will never be another SAT Math question that asks about the border tiles on square game boards.
Instead, the goal of going over this question is to deepen your understanding of the principles of SAT Math in general. There’s no way to predict exactly what kinds of things you’ll see on test day, but if we come to understand the importance of reading carefully, thinking about the answer choices as part of the question, and so on, then you’ll be able to take apart whatever weird combinations of basic facts the SAT presents you with. So it’s not about memorizing rigid steps for certain types of questions. It’s about developing a general feel, and confidence that you can work out whatever they throw at you by relying on the test’s design principles.
Page 518, Question 17
Just about everyone panics for a second when they run into this question, because it seems to be asking us to figure out the area of a shape we’ve never seen before. This is one of those moments when knowing the unwritten rules of the test really comes in handy. Remember that the College Board can only ask you to find the areas of rectangles, triangles, and circles, and it gives you the formulas for those areas at the beginning of each SAT Math section.
So if a question looks like it's asking for the area of something else without giving you another area formula to use, then that something else can always be expressed in terms of rectangles, triangles, and circles. Always.
In this case, it’s probably pretty clear that we can’t use triangles and rectangles, because there are no corners in this figure. So we’ll have to figure out the area of these figures as though they were circles.
But how can we do that?
I think the easiest way is to imagine reversing the bottom half of the big circle from left to right, so that the bottom half of the circle becomes just a reflection of the top half, and we're left with 3 circles, all tangent to each other on their left-most points. It would look something like this:
So now we have to find the areas of those circles, and add and subtract them appropriately. If AD is 6, then each of those dots is 1 unit from the dot before it or after it. That means the radius of the smallest circle is 1, so its area is pi. The radius of the biggest circle is 3 units, so its area is 9pi. The radius of the medium circle, the unshaded one, is 2 units, so its area is 4pi.
So we want the amount equal to the area of the biggest circle minus the area of the medium circle plus the area of the smallest. So it's 9pi – 4pi + 1pi, or 6pi. Which means (C) is correct.
This is just one more example of a question that seems a lot more exotic than it is. If you remember the rules the College Board has to play by, then you’ll find a lot of things much easier than most people will. An untrained test-taker will throw up his hands in frustration over this question, but a trained test-taker knows how to turn it into a simple question about circles, and then solve it using the basic formula provided in the beginning of each SAT Math section.
Page 519, Question 18
For this question, I would just draw 6 points out so that no 3 are on a line together, and then try connecting them: basically, each of the 6 points connects to the other 5. We can count the lines up after drawing them out, or we can try doing a little multiplication.
If we multiply, it might seem like there should be 30 lines, since 6 * 5 = 30, but we have to remember that each line touches two of the points. So there aren’t actually 30 lines, because each line counts as a connection for both of the points. In other words, if we call the points A, B, C, D, E, and F, then the line from A to B is the same line as the one from B to A.
So we want to divide the 30 apparent connections by 2 in order to compensate for the fact that each line serves as one connection between 2 points, so we don’t double-count the lines. That gives us 15 for our final answer, so (A) is correct.
Note the patterns in the answer choices: 15 is half of 30, and there's also 36 and 18 (36 would be 6 x 6 instead of 6 x 5, and 18 is half of that). Once more, the answer choices help point us in the right direction and make us aware of potential mistakes that could be easily made.
Page 519, Question 19
This question often blows people away because it seems much more complicated than it actually is—which, as we’ve said many times, is typical for SAT Math questions in general. As usual, we’ll approach this by reading carefully and thinking carefully.