ACT Math Formulas
Weighted Average
\( \frac{\sum (w_i x_i)}{\sum w_i} \)
Example: Tests (weight 3) 90%, Quizzes (weight 1) 80%: \( \frac{3(90)+1(80)}{4} = 87.5\% \)
Area of Rectangle
\( A = l \times w \)
Example: length 5, width 3: \( A = 5 \times 3 = 15 \)
Area of Triangle
\( A = \frac{1}{2} b h \)
Example: base 8, height 4: \( A = \frac{1}{2} \times 8 \times 4 = 16 \)
Area of Parallelogram
\( A = b \times h \)
Example: base 6, height 3: \( A = 6 \times 3 = 18 \)
Area of Trapezoid
\( A = \frac{1}{2} (b_1 + b_2) h \)
Example: bases 5 & 7, height 4: \( A = \frac{1}{2}(5+7)4 = 24 \)
Circumference of Circle
\( C = 2 \pi r \)
Example: radius 3: \( C = 2\pi(3) \approx 18.85 \)
Area of Circle
\( A = \pi r^2 \)
Example: radius 4: \( A = \pi(16) \approx 50.27 \)
Volume of Rectangular Prism
\( V = l \times w \times h \)
Example: l=5, w=3, h=2: \( V = 5\times3\times2 = 30 \)
Volume of Cone
\( V = \frac{1}{3}\pi r^2 h \)
Example: radius 2, height 6: \( V = \frac{1}{3}\pi(4)(6) = 8\pi \approx 25.13 \)
Volume of Pyramid
\( V = \frac{1}{3} B h \)
Example: base area 9, height 5: \( V = \frac{1}{3}(9)(5) = 15 \)
Quadratic Discriminant
\( D = b^2 - 4ac \)
Example: for \( x^2 - 4x + 3=0 \): \( D = (-4)^2 -4(1)(3)=4>0 \) (two real roots)
Slope-Intercept Form
\( y = mx + b \)
Example: slope 2, intercept -3: \( y = 2x - 3 \)
Standard Form of a Line
\( Ax + By = C \)
Example: \( 2x + 3y = 6 \) can be rewritten as \( y = -\tfrac{2}{3}x + 2 \)
Difference of Squares
\( a^2 - b^2 = (a - b)(a + b) \)
Example: \( 9 - 4 = (3 - 2)(3 + 2) = 5 \)
Algebra Essentials
Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Example: Solve \( 2x^2 - 4x - 6 = 0 \)
Solution: \( x = \frac{4 \pm \sqrt{16 + 48}}{4} = \frac{4 \pm 8}{4} \)
Solution: \( x = \frac{4 \pm \sqrt{16 + 48}}{4} = \frac{4 \pm 8}{4} \)
Geometry Fundamentals
Law of Cosines: \( c^2 = a^2 + b^2 - 2ab\cos(C) \)
Example: Triangle with sides 5, 7, angle 60° between them
\( c^2 = 25 + 49 - 70\cos(60°) = 74 - 35 = 39 \)
\( c^2 = 25 + 49 - 70\cos(60°) = 74 - 35 = 39 \)
Arithmetic Sequence
\( a_n = a_1 + (n-1)d \)
Example: First term 3, common difference 2: \( a_5 = 3 + 2(4) = 11 \)
Geometric Sequence
\( a_n = a_1 \cdot r^{n-1} \)
Example: First term 2, ratio 3: \( a_4 = 2 \cdot 3^{3} = 54 \)
Exponential Growth
\( A = P(1 + r)^t \)
Example: Investment grows 5% annually: \( A = 1000(1.05)^5 \)
Logarithm Change of Base
\( \log_b a = \frac{\log_c a}{\log_c b} \)
Example: \( \log_2 8 = \frac{\log_{10} 8}{\log_{10} 2} = 3 \)
Law of Sines
\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
Example: Given a=7, A=30°, B=45°: \( b = \frac{7 \sin 45°}{\sin 30°} \approx 9.9 \)
Pythagorean Theorem
\( a^2 + b^2 = c^2 \)
Example: Right triangle legs 3 & 4: \( c = \sqrt{3^2 + 4^2} = 5 \)
Distance Formula
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Example: Points (1,2) & (4,6): \( d = \sqrt{3^2 + 4^2} = 5 \)
Midpoint Formula
\( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
Example: Points (1,2) & (3,4): Midpoint = (2, 3)
Volume of Cylinder
\( V = \pi r^2 h \)
Example: Radius 2, height 5: \( V = \pi \times 4 \times 5 \approx 62.83 \)
Surface Area of Sphere
\( A = 4\pi r^2 \)
Example: Radius 3: \( A = 4\pi \times 9 \approx 113.1 \)
Combination Formula
\( C(n, k) = \frac{n!}{k!(n-k)!} \)
Example: Choose 2 from 4: \( C(4, 2) = 6 \)
Sum of Interior Angles
\( (n-2) \times 180° \)
Example: Hexagon (n=6): \( 4 \times 180° = 720° \)
Probability
\( P(A) = \frac{\text{Favorable}}{\text{Total}} \)
Example: Rolling a 3: \( P(3) = \frac{1}{6} \)
Slope Formula
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Example: Points (1,2) & (3,6): \( m = \frac{6-2}{3-1} = 2 \)
Percent Change
\( \text{Percent Change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\% \)
Example: Price increase from $50 ? $65: \( \frac{65-50}{50} \times 100\% = 30\% \)
Simple Interest
\( I = Prt \)
Example: $1000 at 5% for 3 years: \( I = 1000 \times 0.05 \times 3 = \$150 \)
Vertex Form of Parabola
\( y = a(x-h)^2 + k \)
Example: Vertex at (2,5): \( y = 2(x-2)^2 + 5 \)
Permutations
\( P(n, k) = \frac{n!}{(n-k)!} \)
Example: Arrange 5 books with 3 slots: \( P(5,3) = 60 \)
30-60-90 Triangle
Ratio: \( 1 : \sqrt{3} : 2 \)
Example: Short leg = 4 ? hypotenuse = 8, long leg = \( 4\sqrt{3} \)
45-45-90 Triangle
Ratio: \( 1 : 1 : \sqrt{2} \)
Example: Legs = 5 ? hypotenuse = \( 5\sqrt{2} \)
Logarithm Properties
\( \log_b(xy) = \log_b x + \log_b y \)
\( \log_b(x/y) = \log_b x - \log_b y \)
Example: \( \log_2(8x) = \log_2 8 + \log_2 x = 3 + \log_2 x \)
Matrix Multiplication
\( [a\ b] \times [c\ d]^T = ac + bd \)
Example: \( [2\ 3] \times [4\ 5]^T = 2(4) + 3(5) = 23 \)
Trig Identity
\( \sin^2\theta + \cos^2\theta = 1 \)
Example: If \( \sin\theta = 0.6 \), then \( \cos\theta = \sqrt{1 - 0.36} = 0.8 \)
Area of Sector
\( A = \frac{1}{2} r^2 \theta \) (\(\theta\) in radians)
Example: Radius 6, angle ?/3: \( A = \frac{1}{2} \times 36 \times \frac{\pi}{3} = 6\pi \)
Weighted Average
\( \frac{\sum (w_i x_i)}{\sum w_i} \)
Example: Tests (weight 3) 90%, Quizzes (weight 1) 80%: \( \frac{3(90)+1(80)}{4} = 87.5\% \)
Science Strategies
Data Representation
- Skim graphs & tables first
- Identify axes and units
- Look for trends & outliers
Research Summaries
- Highlight hypotheses & methods
- Note sample size & variables
- Understand result meanings
Conflicting Viewpoints
- Compare author opinions
- Map contrasting arguments
- Pick details supporting each side
Table Interpretation
- Check headings & units
- Find specific row/column
- Use quick calculations
Experimental Design Analysis
- Identify control & experimental groups
- Note independent/dependent variables
- Assess validity of conclusions
Scientific Notation & Units
- Convert exponents accurately
- Ensure unit consistency
- Use estimation when possible
Hypothesis Testing
- Identify predicted outcomes
- Compare to actual results
- Select statements matching data
Timing & Pacing
- Allocate ~5 min per passage
- Skip and return to tough questions
- Use process of elimination
Punctuation Mastery
Commas
- To separate items:
apples, oranges, and bananas - Before conjunctions:
She studied hard, but still felt nervous. - With nonessential clauses:
My brother, who loves science, aced the test.
Semicolons
- Linking independent clauses:
He’s good at math; she excels in English. - In complex lists:
We visited Dallas, TX; Austin, TX; and Houston, TX.
Colons
- Before a list:
Bring three things: pencils, paper, and a calculator. - To introduce a quote:
He said it best: “Practice makes perfect.”
Dashes & Parentheses
- For emphasis with dashes:
She won the award — and deservedly so. - For asides with parentheses:
The experiment (which took all day) was a success.
Grammar & Usage
- Subject–Verb Agreement:
The list of items is long. - Pronoun Agreement:
Everyone should bring his or her notebook. - Misplaced Modifiers:
Running fast, the finish line appeared!?Running fast, she saw the finish line.
Sentence Structure
- Fragments vs. Run-Ons:
Because it rained. We stayed inside.?Because it rained, we stayed inside. - Parallel Structure:
She likes hiking, to swim, and biking.?She likes hiking, swimming, and biking.
Punctuation Mastery
- Commas
After dinner, we went home. - Semicolons
She aced math; he nailed English. - Colons
Remember: study, practice, repeat. - Em Dashes
Her plan—though risky—worked. - Apostrophes
It’s time for the student’s award.
Rhetorical Skills & Style
- Active vs. Passive:
They conducted the study.(active) vs.The study was conducted.(passive) - Tone & Diction: Formal (“do not”) vs. informal (“don’t”)
- Conciseness: Remove redundancies (“each and every” ? “each”)
Transitions & Organization
- Contrast: however, on the other hand
- Addition: furthermore, moreover
- Cause & Effect: therefore, as a result
Vocabulary & Context Clues
- Definition Clue: “Photosynthesis, the process by which….”
- Contrast Clue: “Unlike mammals, reptiles…”
- Example Clue: “Celestial bodies—such as stars and planets—…”
Timing & Strategy
- Allocate ~8 min per passage
- Answer easy questions first
- Flag tough ones & revisit