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The 28 Critical SAT Math Formulas You MUST Know

Posted by Courtney Montgomery | May 20, 2015 11:57:37 PM

SAT Math

 

feature_pen_and_formula.jpgThe SAT math test is unlike any math test you’ve taken before. It’s designed to take concepts you’re used to and make you apply them in new (and often strange) ways. It’s tricky, but with attention to detail and knowledge of the basic formulas and concepts covered by the test, you can improve your score. 

So what formulas do you need to have memorized for the SAT math section before the day of the test? In this complete guide, I'll cover every critical formula you MUST know before you sit down for the test. I'll also explain them in case you need to jog your memory about how a formula works. If you understand every formula in this list, you'll save yourself valuable time on the test and probably get a few extra questions correct.

 

Formulas Given on the SAT, Explained

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This is exactly what you'll see at the beginning of both math sections (the calculator and no calculator section). It can be easy to look right past it, so familiarize yourself with the formulas now to avoid wasting time on test day. 

 

You are given 12 formulas on the test itself and three geometry laws. It can be helpful and save you time and effort to memorize the given formulas, but it is ultimately unnecessary, as they are given on every SAT math section. 

You are only given geometry formulas, so prioritize memorizing your algebra and trigonometry formulas before test day (we'll cover these in the next section). You should focus most of your study effort on algebra anyways, because geometry has been de-emphasized on the new SAT and now makes up just 10% (or less) of the questions.

Nonetheless, you do need to know what the given geometry formulas mean. The explanations of those formulas are as follows:

 

Area of a Circle

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$$A=πr^2$$

  • π is a constant that can, for the purposes of the SAT, be written as 3.14 (or 3.14159)
  • r is the radius of the circle (any line drawn from the center point straight to the edge of the circle)

 

Circumference of a Circle

$$C=2πr$$

or $$C=πd$$

  • is the diameter of the circle. It is a line that bisects the circle through the midpoint and touches two ends of the circle on opposite sides. It is twice the radius.

 

Area of a Rectangle

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$$A = lw$$

  • l is the length of the rectangle
  • w is the width of the rectangle

 

Area of a Triangle

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$$A = ½bh$$

  • b is the length of the base of triangle (the edge of one side)
  • h is the height of the triangle
    • In a right triangle, the height is the same as a side of the 90-degree angle. For non-right triangles, the height will drop down through the interior of the triangle, as shown above. 

 

The Pythagorean Theorem

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$$a^2 + b^2 = c^2$$

  • In a right triangle, the two smaller sides (a and b) are each squared. Their sum is the equal to the square of the hypotenuse (c, longest side of the triangle).

 

Properties of Special Right Triangle: Isosceles Triangle

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  • An isosceles triangle has two sides that are equal in length and two equal angles opposite those sides.
  • An isosceles right triangle always has a 90-degree angle and two 45 degree angles.
  • The side lengths are determined by the formula: $x$, $x$, $x√2$, with the hypotenuse (side opposite 90 degrees) having a length of one of the smaller sides *$√2$.
    • E.g., An isosceles right triangle may have side lengths of $12$, $12$, and $12√2$.

 

Properties of Special Right Triangle: 30, 60, 90 Degree Triangle

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  • A 30, 60, 90 triangle describes the degree measures of the triangle's three angles.
  • The side lengths are determined by the formula: $x$, $x√3$, and $2x$
    • The side opposite 30 degrees is the smallest, with a measurement of $x$.
    • The side opposite 60 degrees is the middle length, with a measurement of $x√3$.
    • The side opposite 90 degree is the hypotenuse (longest side), with a length of $2x$.
    • For example, a 30-60-90 triangle may have side lengths of $5$, $5√3$, and $10$.

 

Volume of a Rectangular Solid

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$$V = lwh$$

  • l is the length of one of the sides.
  • h is the height of the figure. 
  • w is the width of one of the sides. 

 

Volume of a Cylinder

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$$V=πr^2h$$

  • $r$ is the radius of the circular side of the cylinder.
  • $h$ is the height of the cylinder. 

 

Volume of a sphere

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$$V=(4/3)πr^3$$

  • $r$ is the radius of the sphere.

 

Volume of a Cone

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$$V=(1/3)πr^2h$$

  • $r$ is the radius of the circular side of the cone.
  • $h$ is the height of the pointed part of the cone (as measured from the center of the circular part of the cone). 

 

 

Volume of a Pyramid

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$$V=(1/3)lwh$$

  • $l$ is the length of one of the edges of the rectangular part of the pyramid.
  • $h$ is the height of the figure at its peak (as measured from the center of the rectangular part of the pyramid). 
  • $w$ is the width of one of the edges of the rectangular part of the pyramid.

 

Law: the number of degrees in a circle is 360

Law: the number of radians in a circle is 2π

Law: the number of degrees in a triangle is 180

 

body_brain_power.jpgGear up that brain because here come the formulas you have to memorize.

 

Formulas Not Given on the Test

For most of the formulas on this list, you'll simply need to buckle down and memorize them (sorry). Some of them, however, can be useful to know but are ultimately unnecessary to memorize, as their results can be calculated via other means. (It's still useful to know these, though, so treat them seriously).

We've broken the list into "Need to Know" and "Good to Know," depending on if you are a formula-loving test taker or a fewer-formulas-the-better kind of test taker. 

 

Slopes and Graphs

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Need to Know

  • Midpoint formula
    • Given two points, $A (x_1, y_1)$,$B (x_2, y_2)$, find the midpoint of the line that connects them:

$$({(x_1 + x_2)}/2, {(y_1 + y_2)}/2)$$

 

  • Slope formula
    • Given two points, $A (x_1, y_1)$,$B (x_2, y_2)$, find the slope of the line that connects them: 

      $$(y_2 - y_1)/(x_2 - x_1)$$

    • The slope of a line is the ${\rise (\vertical \change)}/ {\run (\horizontal \change)}$.


  • How to write the equation of a line
    • The equation of a line is written as: $$y = mx + b$$
      • If you get an equation that is NOT in this form (ex. $mx-y = b$), then re-write it into this format! It is very common for the SAT to give you an equation in a different form and then ask you about whether the slope and intercept are positive or negative. If you don’t re-write the equation into $y = mx + b$, and incorrectly interpret what the slope or intercept is, you will get this question wrong. 
    • m is the slope of the line.
    • b is the y-intercept (the point where the line hits the y-axis).
    • If the line passes through the origin $(0,0)$, the line is written as $y = mx$.

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Good to Know

  • Distance formula
    • Given two points, $A (x_1, y_1)$,$B (x_2, y_2)$, find the distance between them:
$$√[(x_2 - x_1)^2 + (y_2 - y_1)^2]$$
  • You don’t need this formula, as you can simply graph your points and then create a right triangle from them. The distance will be the hypotenuse, which you can find via the pythagorean theorem. 

 

Circles

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Good to Know

  • Circumference of an arc
    • Given a radius and a degree measure of an arc from the center, find the circumference of the arc
    • Use the formula for the circumference multiplied by the angle of the arc divided by the total angle measure of the circle (360)
      • $$C_{\arc} = (2πr)({\degree \measure \center \of \arc}/360)$$
      • E.g., A 60 degree arc is $1/6$ of the total circumference because $60/360 = 1/6$
  • Area of an arc
    • Given a radius and a degree measure of an arc from the center, find the circumference of the arc
      • use the formula for the area multiplied by the angle of the arc divided by the total angle measure of the circle
        • $$A_{\arc} = (πr^2)({\degree \measure \center \of \arc}/360)$$
  • An alternative to memorizing the “formula” is just to stop and think about arc circumferences and arc areas logically.
    • You know the formulas for the area and circumference of a circle (because they are in your given equation box on the test).
    • You know how many degrees are in a circle (because it is in your given equation box on the text).
    • Now put the two together:
      • If the arc spans 90 degrees of the circle, it must be $1/4$th the total area/circumference of the circle because $360/90 = 4$. If the arc is at a 45 degree angle, then it is $1/8$th the circle, because $360/45 = 8$.
      • The concept is exactly the same as the formula, but it may help you to think of it this way instead of as a “formula” to memorize.

 

Averages

Need to Know

  • The average is the same thing as the mean
  • Find the average/mean of a set of numbers/terms
$$\Mean = {\sum \of \the \terms}/{\number \of \different \terms}$$
  •  Find the average speed

$$\Speed = {\total \distance}/{\total \time}$$

 

Probabilities

Need to Know

  • Probability is a representation of the odds of something happening. A probability of 1 is guaranteed to happen. A probability of 0 will never happen.

$$\text"Probability of an outcome" = {\text"number of desired outcomes"}/{\text"total number of possible outcomes"}$$

  •  Probability of two, mutually exclusive, outcomes both happening is

$$\text"Probability of event A" * \text"probability of event B"$$

 

Percentages

Need to know

  • Find x percent of a given number n.

$$n(x/100)$$

  • Find out what percent a number n is of another number m.

$$(n100)/m$$

  • Find out what number n is x percent of.
$$(n100)/x$$ 

 

Trigonometry 

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Trigonometry is a new addition to the new 2016 SAT math section. Though it makes up less than 5% of math questions, you won't be able to answer the trigonometry questions without knowing the following formulas.

 

Need to Know

  • Find the sine of an angle given the measures of the sides of the triangle. 

$sin(x)$= Measure of the opposite side to the angle / Measure of the hypotenuse 

In the figure above, the sine of the labeled angle would be $a/h$.

  • Find the cosine of an angle given the measures of the sides of the triangle. 

$cos(x)$= Measure of the adjacent side to the angle  Measure of the hypotenuse 

In the figure above, the cosine of the labeled angle would be $b/h$.

  • Find the tangent of an angle given the measures of the sides of the triangle. 

$tan(x)$= Measure of the opposite side to the angle Measure of the adjacent side to the angle

In the figure above, the tangent of the labeled angle would be $a/b$.

  • A helpful memory trick is an acronym: SOHCAHTOA. 

Sine equals Opposite over Hypotenuse

Cosine equals Adjacent over Hypotenuse

Tangent equals Opposite over Adjacent 

 

Keep in Mind

Though these are all the formulas you’ll need, both ones you’re given and ones you need to memorize, this list does not cover every aspect of the SAT math. For example, you’ll also need to understand how to factor equations, how to manipulate and solve absolute values, and how to manipulate and use exponents, and much more. These topics are all covered here.

 

What's Next?

Now that you know the critical formulas for the SAT, it might be time to check out the complete list of SAT math knowledge and know-how you'll need before test day. And for those of you with particularly lofty score goals, check out our article on How to an 800 on the SAT Math by a perfect SAT-Scorer. 

Currently scoring in the mid-range? Look no further than our article on how to improve your score if you're currently scoring below the 600 range. 

 

Want to improve your SAT score by 160 points? We have the industry's leading SAT prep program. Built by Harvard grads and SAT full scorers, the program learns your strengths and weaknesses through advanced statistics, then customizes your prep program to you, so you get the most effective prep possible.

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Courtney Montgomery
About the Author

Courtney scored in the 99th percentile on the SAT in high school and went on to graduate from Stanford University with a degree in Cultural and Social Anthropology. She is passionate about bringing education and the tools to succeed to students from all backgrounds and walks of life, as she believes open education is one of the great societal equalizers. She has years of tutoring experience and writes creative works in her free time.



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