prepsactSAT & ACT Mastery
Reference & Cheat Sheet Vault

SAT & ACT Formula & Grammar Vault

Complete catalog of mathematical formulas, geometry identities, and standard English punctuation rules.

Area & Circumference of a Circle

GeometryBOTH
A=πr2,C=2πr=πdA = \pi r^2, \quad C = 2\pi r = \pi d

Calculates the 2D surface area and outer boundary perimeter of a circle of radius $r$.

Example:
If diameter is 10, radius r=5r = 5, Area =25π= 25\pi, Circumference =10π= 10\pi.
Remember: Arc length $= \frac{\theta}{360^\circ} \cdot 2\pi r$ (or $s = r\theta$ in radians).

Volume of a Rectangular Prism & Cylinder

GeometryBOTH
V=ℓwh,V=πr2hV = \ell w h, \quad V = \pi r^2 h

Volume represents the 3D interior capacity of right rectangular prisms and cylinders.

Example:
Cylinder with radius 3 and height 5 has volume V=π(32)(5)=45πV = \pi(3^2)(5) = 45\pi.
Surface area of closed cylinder $= 2\pi r^2 + 2\pi r h$.

Volume of Sphere, Cone & Pyramid

GeometrySAT
Vsphere=43πr3,Vcone=13πr2h,Vpyramid=13ℓwhV_{\text{sphere}} = \frac{4}{3}\pi r^3, \quad V_{\text{cone}} = \frac{1}{3}\pi r^2 h, \quad V_{\text{pyramid}} = \frac{1}{3} \ell w h

Official Digital SAT formula reference for 3D curved and tapering geometric solids.

Example:
A cone with the same base and height as a cylinder has exactly 13\frac{1}{3} the volume.
Provided on the official SAT Math formula reference popup.

Special Right Triangles ($30^\circ-60^\circ-90^\circ$ and $45^\circ-45^\circ-90^\circ$)

GeometryBOTH
30∘−60∘−90∘:x,x3,2x∣45∘−45∘−90∘:x,x,x230^\circ-60^\circ-90^\circ: x, x\sqrt{3}, 2x \quad | \quad 45^\circ-45^\circ-90^\circ: x, x, x\sqrt{2}

Exact side ratio relationships for standard right triangles.

Example:
In a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle with hypotenuse 10: short side =5= 5, long leg =53= 5\sqrt{3}.
Extremely common in SAT Math coordinate geometry and trigonometry questions.

Pythagorean Theorem & Distance Formula

GeometryBOTH
a2+b2=c2,d=(x2−x1)2+(y2−y1)2a^2 + b^2 = c^2, \quad d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Calculates the hypotenuse or Euclidean distance between points $(x_1, y_1)$ and $(x_2, y_2)$.

Example:
Common Pythagorean triples: (3,4,5)(3, 4, 5), (5,12,13)(5, 12, 13), (7,24,25)(7, 24, 25), (8,15,17)(8, 15, 17).
Multiples like $(6, 8, 10)$ and $(10, 24, 26)$ frequently appear.

Linear Equations & Perpendicular Slopes

AlgebraBOTH
y=mx+b,m⊥=−1my = mx + b, \quad m_{\perp} = -\frac{1}{m}

Slope-intercept form where $m$ is slope and $b$ is $y$-intercept. Perpendicular lines have negative reciprocal slopes.

Example:
If line 1 has slope m=23m = \frac{2}{3}, a perpendicular line must have slope m⊥=−32m_\perp = -\frac{3}{2}.
Parallel lines have identical slopes ($m_1 = m_2$) and different $y$-intercepts.

Sum & Product of Quadratic Roots (Vieta's Formulas)

AlgebraBOTH
x1+x2=−ba,x1⋅x2=cax_1 + x_2 = -\frac{b}{a}, \quad x_1 \cdot x_2 = \frac{c}{a}

Find the sum or product of solutions for $ax^2 + bx + c = 0$ without solving the quadratic equation.

Example:
For 3x2−12x+7=03x^2 - 12x + 7 = 0, the sum of solutions is −−123=4-\frac{-12}{3} = 4.
Massive time-saver on the Digital SAT and ACT Math sections!

Exponential Growth & Compound Interest

AlgebraBOTH
A(t)=P(1+r)t,A(t)=P(1+rn)ntA(t) = P(1 + r)^t, \quad A(t) = P\left(1 + \frac{r}{n}\right)^{nt}

Models growth where $P$ is principal, $r$ is decimal rate, and $n$ is compounding frequency per year.

Example:
An investment of \1,000growingat61,000 growing at 6% annually fortyears:years:A(t) = 1000(1.06)^t$.
For half-life decay: $A(t) = P\left(\frac{1}{2}\right)^{\frac{t}{h}}$ where $h$ is half-life duration.

SOH-CAH-TOA & Fundamental Trigonometric Identity

TrigonometryBOTH
sin⁡(θ)=OH, cos⁡(θ)=AH, tan⁡(θ)=OA,sin⁡2(θ)+cos⁡2(θ)=1\sin(\theta) = \frac{O}{H}, \, \cos(\theta) = \frac{A}{H}, \, \tan(\theta) = \frac{O}{A}, \quad \sin^2(\theta) + \cos^2(\theta) = 1

Right-triangle trigonometric ratios and Pythagorean identity.

Example:
Complementary angle theorem: sin⁡(x)=cos⁡(90∘−x)\sin(x) = \cos(90^\circ - x).
In radians: $\sin(x) = \cos\left(\frac{\pi}{2} - x\right)$.

Mean, Margin of Error & Confidence Intervals

StatisticsBOTH
xˉ=∑xin,CI=[Estimate−MoE, Estimate+MoE]\bar{x} = \frac{\sum x_i}{n}, \quad \text{CI} = [\text{Estimate} - \text{MoE}, \, \text{Estimate} + \text{MoE}]

Margin of error decreases as sample size increases, provided the sample is randomly selected.

Example:
A sample mean of 45% with a 3% margin of error indicates a 95% confidence range of 42% to 48%.
Only randomized controlled experiments allow causal conclusions ('X causes Y'). Observational studies only show correlation.

Essential Punctuation Hierarchy (Period > Semicolon > Colon > Dash)

Grammar RulesBOTH
Independent+Independent  ⟹  (.),(;), or (, + FANBOYS)\text{Independent} + \text{Independent} \implies (.), (;), \text{ or } (, \text{ + FANBOYS})

Rules for sentence boundaries and clause joining in SAT/ACT English & Writing.

Example:
Colon (:) requires an independent clause before it: 'She loved one subject: astronomy.'
A pair of em-dashes (— ... —) or parentheses functions identically to non-essential commas.