Formulas to Know for 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).
I'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.
Rectangles
Need to Know
Area of a Rectangle
A = l*w
“l” is the length of the rectangle
“w” is the width of the rectangle
Note: in a square, the length is equal to the width, so you only need to find a single side (A = side^2)
Perimeter of a Rectangle
P = 2*l + 2*w
Triangles
Need to Know
Area of a Triangle
A = 1/2bh
“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, forming a right angle with the base.
The Pythagorean Theorem
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).
Special Right Triangles
45-45-90 Triangle
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.
30-60-90 Triangle
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.
Circles
Need to Know
Area of a Circle
A = πr^2
“r” is the radius of the circle
Circumference of a Circle
C = 2πr (or C = πd)
“d” is the diameter of the circle (d = 2r)
Good to Know
Length of an arc
Given a radius and a degree measure of an arc from the center, find the length 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)
L(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 sector
Given a radius and a degree measure of an arc from the center, find the area of the arc sector
Use the formula for the area multiplied by the angle of the arc divided by the total angle measure of the circle
A(arc sector) = (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 - find what fraction of the whole circle the arc is by degrees and multiply that fraction by the area/circumference of the circle
Slopes and Graphs
Need to Know
Slope Formula
Given two points, A(x1, y1) and B(x2, y2), find the slope of the line that connects them.
Slope = (y2 - y1) / (x2 - x1)
Note: The slope of a line is the rise (vertical change) over the run (horizontal change)
How to write the equation of a line
The equation of a line is written as y = mx + b
Note: if you get an equation that is NOT in this format, rearrange the equation until it is.
“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
Good to Know
Midpoint Formula
Given two points, A(x1, y1) and B(x2, y2), find the slope of the line that connects them.
Midpoint: ((x1 + x2)/2, (y1+y2)/2)
Distance Formula
Given two points, A(x1, y1) and B(x2, y2), find the distance between them.
Distance = sq[(x2-x1)^2 + (y2-y1)^2]
Note: 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.
Algebra
Need to Know
Quadratic Equation
Given a polynomial in the form of ax^2+bx+c, solve for x.
x = [-b +- sq(b^2 - 4ac)] / 2a
Note: memorize this formula to the tune of “Pop Goes the Weasel”
This should only be used if the polynomial is difficult/impossible to factor down
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.
Probability of an outcome = number of desired outcomes / total number of possible outcomes
Good to Know
A probability of 1 is guaranteed to happen. A probability of 0 will never happen.
Percentages
Need to Know
Find x percent of a given number n
Percent = n(x/100)
Find out what percent a number n is of another number m
Percent = (n/m)*100
Trigonometry
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 sine 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 sine of the labeled angle would be a/b
A helpful memory trick is an acronym: SOHCAHTOA
Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent