Formula sheetSAT Math

SAT Math Formula Sheet

Connect algebra, rates, data and geometry formulas to the conditions that make them useful. Each entry includes an original worked example and links to existing lessons and practice.

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Selected SAT Math study formulas across the four current content domains. Official provided-status labels were checked against Fall 2026 Bluebook directions, physical page 8; supplementary formulas are identified separately. This is not a complete syllabus, official reference-sheet reproduction or score guarantee.

Original examples and source-located provided-status checks; independent agent review and qualified educator review are separate. Existing linked practice retains its published foundation scope.

Independent content review: peer checkedQualified educator review: pending

Independent checks and qualified educator review are separate. This original study resource does not claim official endorsement, a validated score prediction or a pass guarantee.

Use the formula after identifying the relationship

This is an independently written study reference, not a copy of the official reference sheet and not a complete list of every fact the SAT can assess. The worked examples below are original teaching examples. Each formula links to existing OpenExam lessons and related practice; those linked questions are additional practice, not the example printed here.

The official Fall 2026 Bluebook directions show the complete reference sheet on physical page 8. We inspected that page to label each entry as provided, supplemental, or a derivation from provided facts. The current assessment framework is Version 4, September 2026. Check the reference tool in your own Bluebook test preview before test day.

Name the requested quantity and its units before substituting numbers.
Check nonzero denominators, real square roots, and whether a triangle is actually right.
Keep exact fractions and multiples of pi until a decimal approximation is requested.
A supplied formula does not choose the right model or establish that its conditions hold.

Algebra and quadratic structure

Equations preserve relationships. Before using a shortcut, identify what can be divided, which solutions are real, and what quantity the question actually asks for.

FORMULA & WORKED EXAMPLE

Solve a linear equation

ax+b=c\quad\Longrightarrow\quad x=\frac{c-b}{a}

Variables

a: nonzero coefficient of x
b and c: real constants
x: unknown real value

Worked example

Solve 5x−7=18.

Add 7 to both sides: 5x=25.
Divide by 5: x=5.
Check: 5(5)−7=18.

Answer: x=5

Lesson: linear Practice 1 linked question

FORMULA & WORKED EXAMPLE

Slope and a line through a point

m=\frac{y_2-y_1}{x_2-x_1},\qquad y-y_1=m(x-x_1),\qquad y=mx+b

Variables

(x1,y1), (x2,y2): distinct points on a nonvertical line
m: change in y per one unit of x
b: y-intercept

Worked example

Find an equation of the line through (2,5) and (6,13).

Compute m=(13−5)/(6−2)=8/4=2.
Use y−5=2(x−2), then simplify to y=2x+1.
Check that both supplied points satisfy the equation.

Answer: y=2x+1

Lesson: slope Lesson: bivariate models Practice 2 linked questions

FORMULA & WORKED EXAMPLE

Products, quotients and powers of powers

a^m a^n=a^{m+n},\qquad \frac{a^m}{a^n}=a^{m-n},\qquad (a^m)^n=a^{mn}

Variables

a: common base
m,n: integer exponents in these rules

Worked example

Simplify (2^3 × 2^4)/2^2.

Combine the numerator to 2^(3+4)=2^7.
Subtract the denominator exponent: 2^(7−2)=2^5.
Evaluate 2^5=32.

Answer: 32

Lesson: exponents Practice 1 linked question

FORMULA & WORKED EXAMPLE

Quadratic formula and real-root check

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},\qquad D=b^2-4ac

Variables

a,b,c: real coefficients with a != 0
D: discriminant
x: candidate root

Worked example

Solve 2x²−7x+3=0.

Identify a=2, b=−7, c=3.
Compute D=(−7)^2−4(2)(3)=25.
x=(7±5)/4, so x=3 or x=1/2.

Answer: x=3 or x=1/2

Lesson: quadratics Practice 2 linked questions

FORMULA & WORKED EXAMPLE

Vertex form and the turning point

y=a(x-h)^2+k,\qquad h=-\frac{b}{2a},\qquad k=f(h)

Variables

a,b,c: coefficients of the quadratic
h: x-coordinate of the vertex
k: y-coordinate of the vertex

Worked example

Find the vertex of f(x)=2x²−8x+1.

Compute h=−(−8)/(2×2)=2.
Compute k=f(2)=8−16+1=−7.
Equivalently, f(x)=2(x−2)^2−7.

Answer: The vertex is (2,−7), a minimum.

Lesson: quadratics Lesson: function transformations Practice 1 linked question

FORMULA & WORKED EXAMPLE

Difference of squares

u^2-v^2=(u-v)(u+v)

Variables

u,v: real expressions

Worked example

Factor x²−16 and solve x²−16=0.

Write 16=4².
Factor x²−4²=(x−4)(x+4).
A product is zero when at least one factor is zero.

Answer: (x−4)(x+4); solutions x=4 and x=−4.

Lesson: quadratics Lesson: polynomial reasoning Practice 2 linked questions

Ratios, percent changes and exponential models

Use a baseline and a consistent time unit. A 15% increase followed by a 15% decrease does not restore the starting value because the second percentage uses a different baseline.

FORMULA & WORKED EXAMPLE

Unit rate and distance

v=\frac{d}{t},\qquad d=vt

Variables

d: distance
t: nonzero elapsed time
v: constant speed or average speed over that interval

Worked example

A trip covers 150 km in 2.5 hours. Find its average speed.

Divide total distance by total time: 150/2.5=60.
Attach the compound unit km per hour.

Answer: 60 km/h

Lesson: ratios Practice 1 linked question

FORMULA & WORKED EXAMPLE

Percent change and a multiplier

r=\frac{\text{new}-\text{old}}{\text{old}},\qquad \text{new}=\text{old}(1+r)

Variables

old: nonzero starting value
new: ending value
r: signed proportional change written as a decimal

Worked example

A price rises from $80 to $92. Find the percentage increase and recover the original price from $92.

r=(92−80)/80=12/80=0.15, or 15%.
The multiplier is 1.15.
Reverse it: 92/1.15=80.

Answer: 15% increase; original price $80.

Lesson: percents Practice 1 linked question

FORMULA & WORKED EXAMPLE

Repeated proportional change

A=P(1+r)^t

Variables

P: initial amount
r: signed change per period as a decimal
t: number of equal periods
A: amount after t periods

Worked example

An original model starts at 1,200 units and grows 4% per year. Find its value after 3 years.

The annual multiplier is 1+0.04=1.04.
A=1200(1.04)^3=1349.8368.
If a whole-unit approximation is requested, round only at the end.

Answer: 1,349.8368 units, approximately 1,350 units.

Lesson: exponential Lesson: percents Practice 2 linked questions

Data summaries, conditional probability and model errors

A denominator is part of the question. Decide whether you are describing all observations or only the observations allowed by a condition. A fitted prediction and an observed result are different quantities.

FORMULA & WORKED EXAMPLE

Mean from values and frequencies

\bar x=\frac{\sum f_i x_i}{\sum f_i}

Variables

xi: numerical value
fi: nonnegative count or weight for xi
sum fi: positive total count or weight

Worked example

A data set contains two 3s and three 7s. Find the mean.

Total value is 2×3+3×7=27.
Total count is 2+3=5.
Mean=27/5=5.4.

Answer: 5.4

Lesson: statistics Lesson: distribution comparisons Practice 2 linked questions

FORMULA & WORKED EXAMPLE

Conditional probability from a count table

P(A\mid B)=\frac{n(A\cap B)}{n(B)}

Variables

A,B: events
n(B): number of observations satisfying B
n(A and B): observations satisfying both

Worked example

Of 30 commuters, 12 cycle. Five of those 12 wear helmets. Among the cyclists, what proportion wear helmets?

Restrict the denominator to the 12 cyclists.
Five satisfy both conditions, so the conditional proportion is 5/12.
5/30 would instead be the joint proportion among all commuters.

Answer: 5/12

Lesson: probability Lesson: two way tables Practice 2 linked questions

FORMULA & WORKED EXAMPLE

Observed value minus prediction

e=y-\widehat y

Variables

y: observed response
y-hat: model-predicted response at the supplied x
e: residual, in response units

Worked example

A fitted model predicts 24 minutes, but the observed time is 29 minutes. Find and interpret the residual.

Compute e=29−24=5 minutes.
The observed time is 5 minutes greater than the prediction.

Answer: Residual +5 minutes.

Lesson: bivariate models Practice 1 linked question

Plane geometry and right triangles

A sketch is not a reason to assume a right angle, equal lengths or similarity. Use the stated relationships and distinguish a slanted side from a perpendicular height.

FORMULA & WORKED EXAMPLE

Areas of a rectangle, triangle and circle

A_{\rm rect}=lw,\qquad A_{\rm tri}=\tfrac12 bh,\qquad A_{\rm circle}=\pi r^2

Variables

l,w: rectangle side lengths
b: triangle base; h: perpendicular height to that base
r: circle radius; A: area

Worked example

Find the areas of an 8 cm by 3 cm rectangle, a triangle with base 10 cm and perpendicular height 7 cm, and a circle of radius 3 cm.

Rectangle: 8×3=24 cm².
Triangle: (1/2)×10×7=35 cm².
Circle: pi×3²=9pi cm².

Answer: 24 cm²; 35 cm²; 9pi cm².

Lesson: geometry Practice 2 linked questions

FORMULA & WORKED EXAMPLE

Circumference of a circle

C=2\pi r=\pi d

Variables

r: radius
d=2r: diameter
C: distance around the circle

Worked example

A circle has diameter 10 cm. Find its circumference.

The radius is 10/2=5 cm.
C=2pi(5)=10pi cm.

Answer: 10pi cm

Lesson: geometry Lesson: circle relationships Practice 2 linked questions

FORMULA & WORKED EXAMPLE

Pythagorean theorem

a^2+b^2=c^2

Variables

a,b: perpendicular legs
c: hypotenuse opposite the right angle

Worked example

A right triangle has legs 9 and 12. Find the hypotenuse.

c²=9²+12²=81+144=225.
c=sqrt(225)=15 because a length is nonnegative.

Answer: 15 length units

Lesson: geometry Lesson: triangle relations Practice 2 linked questions

FORMULA & WORKED EXAMPLE

Special right-triangle side ratios

45^\circ\!:\!45^\circ\!:\!90^\circ:\ s,s,s\sqrt2;\qquad 30^\circ\!:\!60^\circ\!:\!90^\circ:\ x,x\sqrt3,2x

Variables

s: either leg of a 45-45-90 triangle
x: short leg opposite 30 degrees in a 30-60-90 triangle

Worked example

A 45-45-90 triangle has leg 6. A different 30-60-90 triangle has short leg 5. Find their missing sides.

The first has other leg 6 and hypotenuse 6sqrt(2).
The second has long leg 5sqrt(3) and hypotenuse 10.

Answer: First: 6 and 6sqrt(2). Second: 5sqrt(3) and 10.

Lesson: geometry Lesson: triangle relations Practice 2 linked questions

FORMULA & WORKED EXAMPLE

Sine, cosine and tangent in a right triangle

\sin\theta=\frac{o}{h},\qquad \cos\theta=\frac{a}{h},\qquad \tan\theta=\frac{o}{a}

Variables

theta: a chosen acute angle of a right triangle
o: leg opposite theta
a: leg adjacent to theta
h: hypotenuse

Worked example

Relative to an acute angle theta, a right triangle has opposite leg 3, adjacent leg 4 and hypotenuse 5. Find the three ratios.

sin(theta)=3/5.
cos(theta)=4/5.
tan(theta)=3/4.

Answer: sin(theta)=3/5, cos(theta)=4/5, tan(theta)=3/4.

Lesson: geometry Lesson: triangle relations Practice 1 linked question

FORMULA & WORKED EXAMPLE

Similarity and geometric scaling

\frac{L_2}{L_1}=k,\qquad \frac{A_2}{A_1}=k^2,\qquad \frac{V_2}{V_1}=k^3

Variables

k>0: common linear scale factor
L: corresponding length
A: area; V: volume

Worked example

Similar solids have corresponding lengths in a 3:1 ratio. The smaller solid has volume 10 cm³. Find the larger volume.

The linear factor from smaller to larger is k=3.
The volume factor is 3³=27.
The larger volume is 27×10=270 cm³.

Answer: 270 cm³

Lesson: geometry Lesson: solid measures Practice 2 linked questions

Coordinates, circles and angle measures

Use coordinates to recover lengths and centers. For arcs and sectors, first check whether an angle is given in degrees or radians; a radius and a diameter are not interchangeable.

FORMULA & WORKED EXAMPLE

Distance and midpoint in the coordinate plane

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2},\qquad M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

Variables

(x1,y1),(x2,y2): points in a rectangular coordinate system
d: nonnegative distance
M: midpoint

Worked example

Find the distance and midpoint between (−1,2) and (5,10).

The changes are 5−(−1)=6 and 10−2=8.
Distance=sqrt(6²+8²)=sqrt(100)=10.
Midpoint=((−1+5)/2,(2+10)/2)=(2,6).

Answer: Distance 10; midpoint (2,6).

Lesson: geometry Lesson: circle relationships Practice 2 linked questions

FORMULA & WORKED EXAMPLE

Center-radius equation of a circle

(x-h)^2+(y-k)^2=r^2

Variables

(h,k): center
r>0: radius
(x,y): any point on the circle

Worked example

Write the equation of the circle with center (2,−3) and radius 4.

Substitute h=2, k=−3 and r=4.
y−(−3) becomes y+3, and r²=16.

Answer: (x−2)²+(y+3)²=16

Lesson: circle relationships Practice 2 linked questions

FORMULA & WORKED EXAMPLE

Triangle angle sum and degree-radian conversion

\alpha+\beta+\gamma=180^\circ,\qquad \theta_{\rm rad}=\theta_{\rm deg}\frac{\pi}{180}

Variables

alpha,beta,gamma: interior angles of a plane triangle
theta_deg,theta_rad: measures of the same angle

Worked example

A triangle has two angles 40° and 65°. Find the third angle and express a separate 150° angle in radians.

Third angle=180−40−65=75°.
150° × pi/180=5pi/6 radians.

Answer: 75°; 5pi/6 radians.

Lesson: triangle relations Lesson: circle relationships Practice 2 linked questions

FORMULA & WORKED EXAMPLE

Arc length and sector area

s=\frac{\theta}{360^\circ}2\pi r,\qquad A=\frac{\theta}{360^\circ}\pi r^2

Variables

theta: central angle in degrees
r: radius
s: arc length; A: sector area

Worked example

A circle has radius 6 cm and a central angle of 60°. Find the corresponding arc length and sector area.

The sector is 60/360=1/6 of the circle.
Arc length=(1/6)×2pi×6=2pi cm.
Sector area=(1/6)×pi×6²=6pi cm².

Answer: Arc length 2pi cm; sector area 6pi cm².

Lesson: circle relationships Practice 2 linked questions

Volumes: choose a base and a perpendicular height

Use cubic units and distinguish a solid’s vertical height from a slant height. These five volume relationships are on the inspected official sheet. The worksheet examples below use new dimensions and original prompts.

FORMULA & WORKED EXAMPLE

Volume of a rectangular prism

V=lwh

Variables

l,w,h: mutually perpendicular edge lengths
V: volume

Worked example

A rectangular box measures 4 cm by 3 cm by 5 cm. Find its volume.

Multiply the three perpendicular dimensions: 4×3×5=60.
The resulting unit is cm³.

Answer: 60 cm³

Lesson: solid measures Practice 1 linked question

FORMULA & WORKED EXAMPLE

Volume of a right circular cylinder

V=\pi r^2h

Variables

r: base radius
h: perpendicular height
V: volume

Worked example

A right circular cylinder has radius 3 cm and height 8 cm. Find its volume.

Base area=pi×3²=9pi cm².
Volume=9pi×8=72pi cm³.

Answer: 72pi cm³

Lesson: solid measures Practice 1 linked question

FORMULA & WORKED EXAMPLE

Volume of a sphere

V=\frac43\pi r^3

Variables

r: sphere radius
V: volume

Worked example

A sphere has radius 3 cm. Find its volume.

r³=3³=27 cm³.
V=(4/3)pi×27=36pi cm³.

Answer: 36pi cm³

Lesson: solid measures Practice 1 linked question

FORMULA & WORKED EXAMPLE

Volume of a right circular cone

V=\frac13\pi r^2h

Variables

r: circular base radius
h: perpendicular height
V: volume

Worked example

A right circular cone has radius 4 cm and perpendicular height 9 cm. Find its volume.

Base area=16pi cm².
V=(1/3)×16pi×9=48pi cm³.

Answer: 48pi cm³

Lesson: solid measures Practice 1 linked question

FORMULA & WORKED EXAMPLE

Volume of a rectangular-base pyramid

V=\frac13lwh

Variables

l,w: perpendicular side lengths of the rectangular base
h: perpendicular height from apex to base plane
V: volume

Worked example

A pyramid has a 6 cm by 5 cm rectangular base and perpendicular height 9 cm. Find its volume.

Base area=6×5=30 cm².
V=(1/3)×30×9=90 cm³.

Answer: 90 cm³

Lesson: solid measures Practice 1 linked question

Method choice and common errors

Before opening a calculator, write a short equation naming the unknown. If the prompt asks for a derived expression, solving for x may only be an intermediate step. After calculation, check the units, whether the magnitude is plausible and whether a candidate value lies in the original domain.

This reference does not replace instruction on inequalities, systems, functions, sampling design or interpretation of distributions. Those ideas often require reasoning rather than a remembered formula. The linked lessons show the method and its limitations.

Do not cancel across addition, divide by an expression that could be zero, or keep an extraneous root after squaring.
For successive percent changes multiply the factors; do not add signed percentages unless the model warrants it.
For geometry select area, perimeter or volume before calculating, and check whether the requested dimension is radius, diameter or height.
A model’s slope, a residual and evidence of causation answer different questions.

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Sources and review notes

The references establish the stated framework and source basis. Original examples and study tasks are independently authored; reference links do not imply endorsement or permission to reproduce official test items.

College Board: The SAT Math Section
Version / date: Current public page retrieved 2026-10-28
Locator: Math domains: Algebra, Advanced Math, Problem-Solving and Data Analysis, Geometry and Trigonometry; saved as sat-math-scope.html.
SAT School Day Bluebook Test Directions — Fall 2026
Version / date: Fall 2026; title-page version checked
Locator: Physical pages 6–8: Math directions and reference sheet; page 8 visually checked against original PDF.
Assessment Framework for the Digital SAT Suite
Version / date: Version 4, September 2026
Locator: Math content specifications and Appendix B skill/knowledge testing points; used to map learning relationships, not to reproduce test questions.
College Board: What to Expect on SAT Test Day
Version / date: Current public page retrieved 2026-10-28
Locator: The Digital Testing Application: reference sheet and calculator available on Math questions.

Lesson: linear Lesson: slope Lesson: bivariate models Lesson: exponents Lesson: quadratics Lesson: function transformations Lesson: polynomial reasoning Lesson: ratios Lesson: percents Lesson: exponential Lesson: statistics Lesson: distribution comparisons Lesson: probability Lesson: two way tables Lesson: geometry Lesson: circle relationships Lesson: triangle relations Lesson: solid measures

Return to the resource library Check the coverage map

Read this scope first

Use the formula after identifying the relationship

Algebra and quadratic structure

Solve a linear equation

Worked example

Slope and a line through a point

Worked example

Products, quotients and powers of powers

Worked example

Quadratic formula and real-root check

Worked example

Vertex form and the turning point

Worked example

Difference of squares

Worked example

Ratios, percent changes and exponential models

Unit rate and distance

Worked example

Divide a total in a stated ratio

Worked example

Percent change and a multiplier

Worked example

Repeated proportional change

Worked example

Data summaries, conditional probability and model errors

Mean from values and frequencies

Worked example

Conditional probability from a count table

Worked example

Observed value minus prediction

Worked example

Plane geometry and right triangles

Areas of a rectangle, triangle and circle

Worked example

Circumference of a circle

Worked example

Pythagorean theorem

Worked example

Special right-triangle side ratios

Worked example

Sine, cosine and tangent in a right triangle

Worked example

Similarity and geometric scaling

Worked example

Coordinates, circles and angle measures

Distance and midpoint in the coordinate plane

Worked example

Center-radius equation of a circle

Worked example

Triangle angle sum and degree-radian conversion

Worked example

Arc length and sector area

Worked example

Volumes: choose a base and a perpendicular height

Volume of a rectangular prism

Worked example

Volume of a right circular cylinder

Worked example

Volume of a sphere

Worked example

Volume of a right circular cone

Worked example

Volume of a rectangular-base pyramid

Worked example

Method choice and common errors

Download the resource PDFs

Sources and review notes