# 10 ACT Math Formulas YOU MUST KNOW (plus other useful formulas)

Quesbook
Test Tips
May 2nd 2017

Here are 10 must-know ACT Math formulas and some other useful formulas that may come in handy on the Math test. We've also included a PDF version for download.

Taking the ACT with only a few days to study? First and foremost, you should understand that the ACT does not provide a list of basic formulas that will be required to know for the test. Thankfully, we've got a great article that helps you pinpoint what formulas to prioritize when studying. These formulas are categorized as Must-Know, Good-to-Know, and Other. While the Must-Know ACT Math Formulas are of the highest priority, going in knowing all of the formulas listed is in your best interest.

* If at any point you feel like you need a more in-depth tutorial of any of these, then visit these articles written about each specific topic!

# 10 Must-Know ACT Math Formulas

## Set up a percentage fraction

$$\dfrac{part}{whole}=\dfrac{x}{100}$$

This is more of a key concept because it can be applied to a variety of questions. Just rearrange the formula to solve for whatever "x" is.

## Standard linear equation

y = mx + b. Where m is the slope and b is the y intercept.

This formula will certainly be on the test. It may not always be written in this standard form, so just be sure that you understand how to rearrange the equation to make it look like this.

## Slope

$$slope = \dfrac{change\;in\;y}{change\;in\;x}$$   aka   $$slope = \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}$$

This formula is so important we have a whole article just on this! You are going to see this on the test and likely in a variety of ways. YOU MUST KNOW THIS!

P= 2l + 2w

## Area of rectangle

A = l $$\times$$ w

## Volume of cuboid

V = l $$\times$$ w $$\times$$ h

## Circumference of circle

$$C=2\pi \;r$$

## Area of circle

$$A=\pi\;r^{2}$$

## Pythagorean theorem

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

This is one of the most, most, most important formulas to know! It's sure to make any appearance on the test, and knowing this will allow you to solve a variety of questions (see distance of a line segment above).

## SOHCAHTOA

$$sin=\dfrac{length\;of\;opposite}{length\;of\;hypotenuse}$$

$$cos=\dfrac{length\;of\;adjacent}{length\;of\;hypotenuse}$$

$$tan=\dfrac{length\;of\;opposite}{length\;of\;adjacent}$$

Most trigonometry you will see on the test can be answered knowing this.

# 9 Good-to-Know ACT Math Formulas

## Mean (average)

$$Mean = \dfrac{sum\;of\;all\;numbers}{number\;of\;numbers}$$

These questions will appear on the test, and this is a relatively easy formula to memorize.

## Probability

$$P = \dfrac{number\;of\;desired\;outcomes}{total\;number\;of\;outcomes}$$

Try to think in terms of percent fraction to help you remember this one.

## Factorials

! = multiply number by each smaller whole number down to one.

Sometimes (but not always!) they remind you in the question what a factorial is.

## Mid point

$$Mid\;Point = (\dfrac{x_{1}+x_{2}}{2},\dfrac{y_{1}+y_{2}}{2})$$

You will probably see a question dealing with the midpoint of a line. This is listed under "good to know" rather than "need to know" because if you think about it, you are basically taking the average of the x and y coordinates, so thinking about this intuitively might be better than pure memorization.

## Length of a line segment

$$L = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$$

This is good to know, but if you are crunched for time studying you can skip memorizing this and simply graph the coordinates; set up a right triangle and solve for the hypotenuse to get the answer. The downside with this method, however, is that it will take longer to answer this type of question come test time!

## Area of a triangle

$$A=\dfrac{1}{2}bh$$

## Surface area of cuboid

TSA = 2(lw + wh + hl)

## Volume of a sphere

$$V=\dfrac{4}{3}\;\pi\;r^{3}$$

## Special triangles

• 3/4/5

A triangle with sides of 3, 4, and 5 will always be a right triangle, even if the problem does not explicitly state it.

• $$45^{\circ}/45^{\circ}/90^{\circ}$$

In these triangles, the ratio of sides will always be 1:1:$$\sqrt{2}$$, with $$\sqrt{2}$$ corresponding to the hypotenuse.

• $$30^{\circ}/60^{\circ}/90^{\circ}$$

In these triangles, the ratio of sides will always be 1:2:$$\sqrt{3}$$. With "2" corresponding to the hypotenuse.

* Knowing these shortcuts will help you save time on the test!

# Some Other Formulas to Keep in Mind

## Geometric and arithmetic sequences

Arithmetic: $$a_{n}$$ = $$a_{1}$$ + d(n-1). Where $$a_{n}$$ is the nth term of the series, $$a_{1}$$ is the first number of the series, and d is the difference between the numbers.

Geometric: $$a_{n}$$ = $$a_{1}$$ x $$r^{n-1}$$. Where $$a_{n}$$ is the nth term of the series, $$a_{1}$$ is the first term in the series, and r is the factor that you multiply by in order to progress from one term to the next.

If you are aiming to get every point possible, more difficult series questions will appear toward the end of the test that would benefit from these formulas.

## Permutations and combinations

Permutations: $$\dfrac{n!}{(n-r)!}$$

Combinations: $$\dfrac{n!}{r!(n-r)!}$$

Again, like the series formulas, questions requiring these formulas will only account for a point or two on the test.

$$x=\dfrac{-b\;+\;\sqrt{b^{2}-4ac}}{2a}$$

This is of lower priority than other formulas because most of the time you can factor the quadratic. However, if a question asks about imaginary numbers with quadratics, then knowing this equation is essential to solving the problem.

## Equation of a circle

$$(x-h)^{2}+(y-k)^{2}=r^{2}$$. Where (h,k) are the center coordinates of the circle.

Usually there is only one question on the test that requires this formula.

## Area of a trapezoid/parallelogram

$$A=\dfrac{a+b}{2}h$$

A trick here is to make triangles and solve the "pieces" of the trapezoid.

## Volumes cones and cylinders

Cone:  $$Volume = \pi ^{2} \dfrac{h}{3}$$

Cylinder: $$Volume = \pi \;r^{2}\;h$$

## Length of chord

chord length = 2rsin($$\dfrac{1}{2}$$$$\theta$$)

## Area of sector

$$A= \pi r^{2} \times \dfrac{\theta}{2\pi }$$

## Law of cos and sine

$$c ^{2}=a^{2}+b^{2}-2ab\;cosC$$

$$\dfrac{a}{sinA}=\dfrac{b}{sinB}=\dfrac{c}{sinC}$$

If you see a trigonometry question that can't be answered knowing SOHCAHTOA, then you will probably need to know these formulas, but they don't come up often.