Are you looking for a quick way to boost your SAT® math score? Well, you’re in luck! These handy formulas are sure to speed up your math problem throughput and help you out when you’re in a tricky spot!
I don’t really have three formulas for you today, but rather three categories of formulas that you really should be familiar with if you are shooting for a high SAT® math score. The three categories of formulas you must know for the SAT® math sections are:
1. Circle sector formulas
2. Vertex Formulas
3. The formulas at the beginning of the test
Without further to do, lets get into the math!
1. Circle sector formulas
First, lets remind ourselves of what exactly a circle sector is. Basically, it is a section of the circle, which you get by sweeping the radius around the circle by a given angle measure. In other words, if you think about cutting a pie, each piece is a sector of the circular pie.
There are two main measurements you will be asked to find: the arc length of a sector, and the area of a sector. The main difference between these two measurements is that arc length is a piece of the circumference of the circle (one dimensional), while area of a sector is a piece of the area of the circle (two dimensional). Remember that these calculations can be done in both degrees and radians (NOT the same formulas!).
Arc Length:
It is very easy to think about these calculations with ratios, since there are only 360° in a circle.
For example, lets say we want the arc length of the sector of half of a circle of radius 5. We know that the entire circumference is given by 2πr, or 10π in this case. To find half of this length, we can simply multiply by 10π by 1/2 to get 5π.
Let’s remember that half of a circle is swept out by 180°. The portion of the circle we are looking at is given by our angle measure divided by the total angle measure of the circle. Therefore, we can rewrite the expression above as 10π times 180°/360°. Even better, this works for any angle, not just nice neat ones like 180°.
This gives us the general formula for arc length in degrees: 2πr • Θ/360°
When working in radians, this formula is basically the same but takes a slightly different form due to the fact that there are 2π radians in a circle.
2πr • Θ(radians)/2π (notice that the 2π cancels) → r • Θ
So for an angle with measure of 1 radian, the arc length is just the length of the radius.
Sector Area:
Once again we use the idea of ratios to make this simple. Instead of multiplying by the total circumference of the circle, this time we multiple by the area of the entire circle, given by the formula πr².
Remember, the portion of the circle we are looking at is given by our angle measure divided by the total angle measure of the circle. Just be sure to remember which angle unit you are using.
This gives us the general formula for sector area in degrees: πr² • Θ/360°
Once again, switching to radians gives us a slightly different form:
πr² • Θ/2π (notice that the π cancels) → (r² • Θ)/2
Be sure to watch the video above for visual examples!
If you are still confused, let me just say that the best way to master these new topics is practice, practice, practice. If you head over the Khan Academy, you can get lots of free practice with these kinds of problems. At the link above, the relevant sections are titled “Angles, arc lengths, and trig functions” and “circle theorems.”
2. Vertex Formulas
There are a few different approaches you can use to find the vertex of a second order polynomial, as long as it is in the standard form y = ax² + bx + c.
1. The first formula is a quick and easy way to find the x coordinate of the vertex given this standard form. The formula is: x = -b/2a
Then, to find the y coordinate, just plug your x value and and you’re good to go.
2. The second formula is more of an analytic approach, involving the knowledge that the vertex sits right in between the zeroes of the function (or between any two points with the same y value for that matter).
For example, if you find that a quadratic equation has zeros at -1 and 5, the vertex must be located on the line x = 2, since this is the average of the zeros and the line of symmetry of the graph.
3. The last way you can go about finding the vertex of a quadratic is by writing it in the standard form f(x) = a(x – h)² + k.
In this form, the vertex is simply given by the point (h,k). Nice and easy!
There are many ways that these formulas can be applied in SAT® math problems, so make sure to hop on Khan Academy and practice!
3. All of the formulas at the beginning of the test
What? You’re joking right? Nope, many people underestimate the importance of these formulas, and for that reason I think it’s a good idea for you guys to memorize them. These formulas include areas of solids, angle identities, and little factoids that just might save you if you’re in a tight spot.
The thing is, since these formulas are given to you, the college board will expect you to be able to use them, and they are likely to make good use of them throughout the test. Being familiar with them can only help you, saving you time and preventing silly mistakes.
If you make some time to brush up on these few formulas before the test, you will be in great shape to kill the SAT® math section! Happy studying!
