Every Math Formula On The SAT
When you go in to take your official SAT on test day, knowing which SAT math formulas are available to you will be an important part of earning a high SAT score. This article breaks down the math formulas given on the test with easy explanations for how to use them. We will also dive into math formulas worth memorizing and understanding that won’t appear on your formula sheet.
Formulas Given On The SAT
The SAT math formula sheet includes 15 key pieces of information for answering math questions:
Let’s break down each of these formulas.
Area And Circumference Of A Circle
The formula for the area (A) of a circle is A = πr2
- Here, r stands for radius, or ½ the length of the circle’s diameter.
- To use this formula, simply square the numerical value of your radius and then multiply this number by π (pi).
The formula for the circumference (C) of a circle is C = 2πr
- To use this formula, multiply the numerical value of your radius by 2, and then multiply that number by π (pi).
Area Of A Rectangle
The formula for the area (A) of a rectangle is A = lw
- Here, l stands for the length of the rectangle and w stands for width of the rectangle.
- To use this formula, multiply the numerical values for the length and width of your rectangle. (This also works for squares!)
Area Of A Triangle
The formula for the area (A) of a triangle is A = ½bh
- Here, b stands for the base of the triangle and h stands for height of the triangle.
- To use this formula, multiply the base (the horizontal line) of the triangle by the height (a straight vertical line from the base to the vertex) of the triangle. Then divide by 2.
The Pythagorean Theorem
The relationship between the three sides of a right triangle can be described using the pythagorean theorem, or a2 + b2 = c2.
- To use this theorem, you only need to know the value of two of the three sides of a right triangle. Say you have a triangle with a base of 3 and a height of 4, but you don’t know the hypotenuse.
- Using the pythagorean theorem, you can set up the equation 32 + 42 = c2
- Solving the problem, you determine that 9 + 16 = c2 , or 25 = c2
- By taking the square root of both sides, you discover that your hypotenuse has a length of 5.
Special Right Triangles
These formulas explain the relationship between angle values and side lengths on right triangles with specific angle measurements.
For a triangle with angle measurements of 90 degrees, 60 degrees, and 30 degrees, there is a consistent relationship between the three side lengths. In this case, the side opposite the 30 degree angle is x, the hypotenuse is 2x, and the final final side is x√3.
- To use this relationship, you only need to know the value of one of the sides to determine the value of the other two. If the hypotenuse is 6, then you know that your x value is 3. The sum of the sides will therefore be x + 2x + x√3, or (3) + (6) + (3√3).
For a triangle with angle measurements of 90 degrees, 45 degrees, and 45 degrees, there is another consistent relationship between the three side lengths. In this case, the hypotenuse has a length of s√2, and the other two sides both have a length of s.
- To use this relationship, you again only need to know the value of one of the sides. If the base and height (the s sides) are 5, the sum of the sides will therefore be s + s + s√2, or (5) + (5) + (5√2).
Volume Of A Rectangular Prism
The formula for the volume (V) of a rectangular prism is V = lwh
- Here, l stands for the length of the prism, w stands for width of the prism, and h stands for the height of the prism.
- To use this formula, simply multiply the numerical values for the length, width, and height of the rectangular prism. (This also works for cubes!)
Note: a rectangular prism can also be called a rectangular solid.
Volume Of A Cylinder
The formula for the volume (V) of a cylinder is V = πr2h
- Here, the r again stands for radius, or ½ the length of the circle’s diameter. The h stands for the height of the cylinder.
- To use this formula, square the numerical value of your radius and then multiply this number by π (pi). Lastly, multiply by the height of the cylinder.
Volume Of A Sphere
The formula for the volume (V) of a sphere is V = 4/3πr3
- To use this formula, cube the numerical value of your radius and then multiply this number by π (pi). Lastly, multiply by 4/3.
Volume Of A Cone
The formula for the volume (V) of a cone is V = 1/3πr2h
- This formula involves four distinct steps. Find your radius and square it, and then multiply that number by π (pi), by the height of the triangular prism, and by ⅓.
Volume Of A Triangular Prism
The formula for the volume (V) of a triangular prism is V = 1/3lwh
- This formula also involves four distinct steps. Find your length (l), width (w), and height. Multiply these three numbers, and then multiply by ⅓.
Three Key Rules
The SAT formula chart also provides three key relationships for solving geometry problems.
- The number of degrees of arc in a circle is 360.
- The number of radians of arc in a circle is 2π.
- The sum of the measures in degrees of the angles of a triangle is 180.
So how do you use these rules?
- When you’re doing a problem that involves the angles in a circle, know that your total angles should always add up to 360 degrees.
- Likewise, the number of radians in a circle should always be 2π.
- Any time you’re solving a triangle problem, the total degrees of the triangle’s three angles should add up to 180.
- EXTRA TIP: To calculate the sum of interior angles for any shape, use the formula (n-2) x 180, where n = the number of sides. If your shape has 7 sides, your formula is (7-2) x 180, or 5 x 180 = 900 degrees!
Formulas Not Given On The SAT
While the formulas given on the SAT math sections are useful for high school geometry problems, there are many other math formulas you need to know on test day. Below, find some of the most important formulas not provided on the SAT formula sheet.
Understanding how to solve problems that involve exponents will be crucial on the SAT. Use the chart below to help!
|Zero Exponent Rule||a0 = 1|
|Product Rule||ax x ay = ax+y|
|Quotient Rule||ax ÷ ay = ax-y|
|Power Rule||(ax)y = axy|
|Negative Exponent Rule||a-x = 1/ax|
The Quadratic Formula
The quadratic formula is used when you’re finding potential x values for a graphing problem involving a parabola.
The formula helps you find your zeroes when your quadratic equation looks like this: ax2 + bx + c = 0.
- To use this formula, plug in your a, b, and c values. Make sure to take the opposite of b, or negative b.
- You’ll solve the equation once using -b + √(b2 – 4ac), and once using -b – √(b2 – 4ac). Adding the first time and subtracting the second time will allow you to find the two values for x.
The Distance Formula
The distance formula is used when determining the distance between two points on a graph. To solve the equation, you’ll need two distinct coordinate points, (x1, y1) and (x2, y2).
- To use this formula, plug in the x and y values of your two distinct points on the graph.
The Midpoint Formula
The midpoint formula is used to find the middle point of a line segment made by two distinct points on a graph. As with the distance formula, you’ll need two distinct coordinate points, (x1, y1) and (x2, y2), to use this equation.
- To find the (x,y) point at the middle of the segment made by your two points, plug in your x1 and x2 and y1 and y2 values.
Equation of a Line
The equation of a line in slope-intercept form looks like this: y = mx + b.
Here, the m value indicates the slope of the line, whereas the b value tells us the y-intercept, or where the line is crossing over the vertical y-axis.
- To use this formula, calculate the slope of your line by finding the rise over the run, or the vertical increase divided by the horizontal increase.
- If we have a line with points at (0, 4) and (1, 6), we have a slope of 2/1, which simplifies to 2. We also see that our y-intercept is at (0,4).
- Therefore, the equation of our line would be y = 2x + 4.
NOTE: Lines can have a positive or negative slope, depending on if they are pointing “up” (y values increasing from left to right) or “down” (y values decreasing from left to right). To determine if your line will be positive or negative, return to the m value. If your line is y = 2x + 4, your graph has a positive slope. If your line is y = -2x + 4, your slope is negative.
Parallel And Perpendicular Lines
Now that we know the equation of a line (y = mx + b), we can think about the relationships between different lines.
- Parallel lines have the same slope but different y-intercepts.
- For example, the line y = 5x + 9 is parallel to the line y = 5x – 14. The two lines will never intersect.
- When perpendicular lines intersect, they form a perfect right angle. These lines have opposite reciprocal slopes. This means the slope is both negated (made negative/positive, depending on the first line) and made reciprocal by swapping the numerator and the denominator.
- For example, if the equation of one line is y = 3/4x + b, the equation of its perpendicular line would be y = -4/3x + b.
- If the equation of one line is y = -2x + b, its perpendicular line would have an equation of y = 1/2x + b.
High School Trigonometry & SOHCAHTOA
On the SAT, you’ll be asked to understand three key elements of trigonometry: sine, cosine, and tangent.
- Calculating sin(x): take angle x and find the side that is opposite to the angle. Divide that side length by the length of the hypotenuse.
- Calculating cos(x): take angle x and find the side that is adjacent to the angle. Divide that sidelength by the length of the hypotenuse.
- Calculating tan(x): take angle x and find the side that is opposite to the angle. Divide that side length by the length of the adjacent side.
How can I remember all this in an easy way? Use the memory trick SOHCAHTOA to remember these relationships.
- SOH – sine = opposite over hypotenuse
- CAH– cosine = adjacent over hypotenuse
- TOA – tangent = opposite over adjacent
Percentages And Ratios
Percentages and ratios are helpful for determining quantity, and often pop up in word problems (finding a value like average speed) on the SAT. Here are the basics:
- To find a percentage x of a number (N), divide x by 100 and multiply by N.
- For example, if my number is 16 and I want to find 75%, I would divide 75/100 and multiply by 16 to find the answer.
- Setting up ratios is an important way to find consistent relationships. This can be used for similar triangle problems, world problems, and basic equations.
- For example, let’s say that I can write 500 words in 60 minutes. How many words can I write in 90 minutes? To solve this problem, set up the ratio 500/60 = x/90, and then cross multiply. 500 x 90 = 4500. Dividing by 60 tells us that I can write 750 words in 90 minutes.
Additional Topics and Advanced Math
On the SAT math test, there are a few tricky (or simply less common) concepts that high school students don’t spend as much time on in school, but may appear in 1-2 questions.
Difference of Squares
The difference of squares formula is this: a2 – b2 = (a + b)(a – b).
- Normal foiling of (a + b)(a – b) results in a2 + ab – ab – b2
- The positive ab and negative ab cancel, leaving you with a2 – b2 – (a + b)(a – b) is simply its factored form.
While you won’t be expected to memorize the standard deviation formula in your test prep, understanding standard deviation may be useful when interpreting graphs and charts on the SAT.
- Standard deviation is ultimately a measure of variation in a set of data. A high standard deviation means that many of the values are close to the average (or mean) of the data set. A low standard deviation means that the values fall on a wider range.
Imaginary and Complex Numbers
Although it makes up a small portion of the SAT, understanding imaginary numbers and complex numbers may be essential for 1-2 problems per test.
An imaginary number is a representation of the square root of a negative number. This is usually represented by the italicized i. The chart below includes the first 8 imaginary numbers and what they are equivalent to, but note that you only have to memorize i, i2, i3, and i4 – after that, the pattern stays consistent.
|i = √-1||i2 = -1||i3 = -i||i4 = 1|
|i5 = √-1||i6 = -1||i7 = -i||i8 = 1|
- To use imaginary numbers, you’ll simply solve equations as you would usually, but swap in the appropriate value for i.
A complex number is just a real number and an imaginary number added together. For example, 3 + 2i is a complex number.
- To solve equations featuring complex numbers, the same rule is true – solve as you would usually, but swap real numbers when possible. If the equation was (3 + 2i) x (3 + 3i), you would solve by foiling to get 9 + 6i + 6i + 6i2. The equation could then be simplified to 9 + 12i + 6(-1), and completed to 3 + 12i.
Arc Length And Arc Area
To find the length of an arc or the area of an arc sector, the degree measure of the arc divided by 360 will be necessary.
- For length of an arc, multiply (2πr)(degree of the arc/360).
- For the area of an arc sector, multiply (πr2)(degree of the arc/360).
Absolute Value Problems
An absolute value problem features a variable inside of ‘toothpicks’ and looks something like this: |x+3| = 12.
- This means that the equation has two solutions: x = -15 and x = 9. When you take the absolute value (or the positive value) of both 12 and -12, the answer is positive 12 and the equation is true.
Equation For A Circle
The equation for a circle reads (x – h)2 + (y – k)2 = r2, where (h,k) is the center point of the circle and r is the circle’s radius.
- This equation can be used to find the center or find the circle’s radius.
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About the Author
Renée Flory is a student at Johns Hopkins University. She studies professional & creative writing and English literature, and she writes short songs & long novels in her free time.