Home β†’ SAT β†’ Formula Blueprint

The Ultimate SAT Math Formula Blueprint

Every formula the SAT can throw at you β€” one card each, with a real example and a diagram where it helps. Start here before past papers.

Geometry Percentages & Interest Rates & Motion Exponents & Radicals Linear Equations Quadratics Coordinate Geometry Statistics Angles & Polygons Trigonometry & Circles Function Transformations
01 Geometry & 3D Volumes
Circles

Area of a Circle

A = Ο€rΒ²
  • r is the radius β€” half the diameter.
  • Square r before multiplying by Ο€.
  • Answers on the SAT almost always stay in terms of Ο€ (e.g. 25Ο€, not 78.5).
r shaded = area
SAT Example
A circle has radius r = 7. Its area is A = Ο€(7)Β² = 49Ο€.
Circles

Circumference of a Circle

C = 2Ο€r = Ο€d
  • Use 2Ο€r if given the radius; use Ο€d if given the diameter.
  • Wheel problems: one full rotation = one circumference.
r dashed line = C
SAT Example
A wheel with radius 4 travels 2Ο€(4) = 8Ο€ per full rotation.
Polygons

Area of a Rectangle

A = l Γ— w
  • l = length, w = width.
  • Works for squares too (where l = w).
l w
SAT Example
A rectangle is 9 long and 4 wide. Area = 9 Γ— 4 = 36.
Triangles

Area of a Triangle

A = Β½bh
  • b = base, h = height. Height must be perpendicular to the base.
  • Works for acute, obtuse, and right triangles β€” the dashed height may fall outside the triangle for obtuse ones.
b h
SAT Example
Base = 10, height = 6. Area = Β½ Γ— 10 Γ— 6 = 30.
Triangles β€” High Frequency

Pythagorean Theorem

aΒ² + bΒ² = cΒ²
  • Only works for right triangles. c is always the hypotenuse (longest side, opposite the right angle).
  • Common Pythagorean triples to memorise: 3-4-5, 5-12-13, 8-15-17.
b a c
SAT Example
Legs a = 6, b = 8. Hypotenuse: c = √(36+64) = √100 = 10. (This is a 3-4-5 triple scaled by 2.)
Triangles β€” High Frequency

Special Right Triangles

45-45-90 & 30-60-90
  • 45Β°-45Β°-90Β°: sides scale as x : x : x√2. If a leg = 5, hypotenuse = 5√2.
  • 30Β°-60Β°-90Β°: sides scale as x : x√3 : 2x. Shortest side opposite 30Β°.
45° 45° x√2 x x 60° 30° 2x x√3 x
SAT Example
30-60-90: shortest side = 4. So x = 4, middle side = 4√3, hypotenuse = 8.
3D Volumes

Volume of a Box (Rectangular Prism)

V = lwh
  • Length Γ— width Γ— height. All three dimensions.
  • Also used for finding the volume of a room, tank, or box in word problems.
SAT Example
A box is 3 Γ— 4 Γ— 5. Volume = 3 Γ— 4 Γ— 5 = 60 cubic units.
3D Volumes

Volume of a Cylinder

V = Ο€rΒ²h
  • Same as (area of circular base) Γ— height.
  • r = radius of the circular cross-section, h = height of the cylinder.
SAT Example
A cylinder with r = 3, h = 5: V = Ο€(9)(5) = 45Ο€.
3D Volumes

Volume of a Sphere

V = 4/3 Β· Ο€rΒ³
  • rΒ³ means radius cubed β€” don't mix up with the circle area formula (rΒ²).
  • Given on the SAT reference sheet, so no need to memorise cold.
SAT Example
Sphere with r = 3: V = 4/3 Γ— Ο€ Γ— 27 = 36Ο€.
3D Volumes

Volume of a Cone

V = β…“Ο€rΒ²h
  • A cone holds exactly β…“ as much as a cylinder with the same base and height.
  • Given on the SAT reference sheet.
SAT Example
Cone with r = 3, h = 4: V = β…“Ο€(9)(4) = 12Ο€.
Polygons

Area of a Trapezoid

A = Β½(b₁ + bβ‚‚)h
  • b₁ and bβ‚‚ are the two parallel sides (bases). h is the perpendicular height between them.
SAT Example
Parallel sides of 6 and 10, height 4: A = Β½(16)(4) = 32.
02 Percentages & Interest
Percentages β€” Core

Percent Formula

Part / Whole = % / 100
  • Cross-multiply to solve for whichever value is missing.
  • Rearranged: Part = Whole Γ— (% / 100).
SAT Example
What is 35% of 80? Part = 80 Γ— 0.35 = 28.
Percentages β€” High Frequency

Percent Change

[(New βˆ’ Old) / Old] Γ— 100
  • Always divide by the original (old) value, not the new one.
  • Positive result = increase. Negative result = decrease.
SAT Example
Price goes from $40 to $52. Change = (52βˆ’40)/40 Γ— 100 = 30% increase.
Percentages β€” Shortcut

Multiplier Method (Increase/Decrease)

Increase: Γ— (1 + r)   Decrease: Γ— (1 βˆ’ r)
  • Convert percentage to a decimal first: 20% β†’ r = 0.20.
  • Faster than computing the percentage separately and adding. Chaining: two 10% increases = Γ— 1.1 Γ— 1.1 = Γ— 1.21 (not 1.20).
SAT Example
A shirt costs $80, discounted 15%. New price: 80 Γ— 0.85 = $68.
Financial β€” Medium Frequency

Simple Interest

A = P(1 + rt)
  • P = principal, r = annual rate (as a decimal), t = time in years.
  • Interest grows at a flat, constant amount each year β€” linear growth.
SAT Example
$1,000 at 5% simple interest for 3 years: A = 1000(1 + 0.05 Γ— 3) = $1,150.
Financial β€” High Frequency

Compound Interest (Annual)

A = P(1 + r)α΅—
  • Interest is added to the principal each year, so next year's interest is on a larger amount β€” exponential growth.
  • Same form as exponential growth generally: y = a(b)Λ£.
SAT Example
$500 at 4% annual compound interest for 2 years: A = 500(1.04)Β² = 500 Γ— 1.0816 = $540.80.
Financial β€” Advanced

Compound Interest (n times/year)

A = P(1 + r/n)^(nt)
  • n = number of compounding periods per year (monthly = 12, quarterly = 4, daily = 365).
  • As n increases, growth approaches continuous compounding A = PeΚ³α΅— β€” but that rarely appears on the SAT.
SAT Example
$1,000 at 6% compounded quarterly for 1 year: A = 1000(1 + 0.06/4)⁴ = 1000(1.015)⁴ β‰ˆ $1,061.36.
03 Rates, Conversions & Motion
Motion β€” High Frequency

Distance, Rate, Time

d = rt
  • d = distance, r = rate (speed), t = time. Rearrange: r = d/t, t = d/r.
  • Units must match: if speed is in mph, time must be in hours.
  • For two objects meeting: add their speeds if travelling toward each other, subtract if same direction.
SAT Example
A car travels 240 miles at 60 mph. Time = d/r = 240/60 = 4 hours.
Rates β€” Accuracy Tool

Unit Conversions (Dimensional Analysis)

Value Γ— (target unit / source unit)
  • Multiply by a fraction equal to 1 (same value, different units) so the original unit cancels.
  • Chain multiple fractions for multi-step conversions.
SAT Example
Convert 3 hours to minutes: 3 hrs Γ— (60 min / 1 hr) = 180 minutes. The "hrs" cancels out.
Mixture Problems

Concentration / Mixture

C₁V₁ + Cβ‚‚Vβ‚‚ = C_final Γ— V_total
  • C = concentration (or strength), V = volume (or amount).
  • The total amount of the substance stays constant β€” mixing doesn't create or destroy it.
SAT Example
Mix 2L of 30% juice with 3L of 50% juice. Final concentration: (0.3Γ—2 + 0.5Γ—3) / 5 = 2.1/5 = 42%.
04 Exponents & Radicals
Exponents β€” Core Rules

Exponent Laws

aᡐ·aⁿ = aᡐ⁺ⁿ   aᡐ/aⁿ = aᡐ⁻ⁿ   (aᡐ)ⁿ = aᡐⁿ
  • Product rule: Same base, multiplying β†’ add the exponents.
  • Quotient rule: Same base, dividing β†’ subtract the exponents.
  • Power rule: Power raised to a power β†’ multiply the exponents.
  • Zero exponent: Any nonzero base to the power 0 = 1.
  • Negative exponent: a⁻ⁿ = 1/aⁿ.
SAT Example
Simplify xΒ³ Β· x⁴: add exponents β†’ x⁷. Simplify x⁢/xΒ²: subtract β†’ x⁴.
Radicals

Fractional Exponents & Radicals

a^(m/n) = ⁿ√(aᡐ)
  • The denominator of the fractional exponent is the root. The numerator is the power.
  • x^(1/2) = √x, x^(1/3) = βˆ›x, x^(2/3) = (βˆ›x)Β².
SAT Example
Simplify 8^(2/3): cube root of 8 is 2, then squared = 4.
05 Linear Equations & Slope
Lines β€” High Frequency

Slope-Intercept Form

y = mx + b
  • m = slope (rise/run). b = y-intercept (where line crosses y-axis).
  • Parallel lines have equal slopes. Perpendicular lines: slopes multiply to βˆ’1.
b run rise
SAT Example
Line y = 2x + 3: slope = 2 (rises 2 for every 1 right), crosses y-axis at (0, 3).
Lines

Point-Slope Form

y βˆ’ y₁ = m(x βˆ’ x₁)
  • Use when you know the slope and one point on the line.
  • Rearrange to slope-intercept form to read off b.
SAT Example
Line through (2, 5) with slope 3: y βˆ’ 5 = 3(x βˆ’ 2) β†’ y = 3x βˆ’ 1.
Lines β€” Core

Slope Formula

m = (yβ‚‚ βˆ’ y₁) / (xβ‚‚ βˆ’ x₁)
  • Rise over run. Order matters β€” be consistent with which point is "1" and which is "2".
  • Horizontal line: slope = 0. Vertical line: slope = undefined.
SAT Example
Points (1, 2) and (4, 8): slope = (8βˆ’2)/(4βˆ’1) = 6/3 = 2.
06 Quadratics
Quadratics β€” High Frequency

Quadratic Formula

x = (βˆ’b Β± √(bΒ²βˆ’4ac)) / 2a
  • Solves axΒ² + bx + c = 0 for any quadratic.
  • Discriminant bΒ²βˆ’4ac: positive = 2 real solutions, zero = 1 solution, negative = no real solutions.
x₁ xβ‚‚ vertex
SAT Example
Solve xΒ² βˆ’ 5x + 6 = 0: a=1,b=βˆ’5,c=6. x = (5 Β± √(25βˆ’24))/2 = (5 Β± 1)/2 β†’ x = 3 or x = 2.
Quadratics

Vertex Form

y = a(x βˆ’ h)Β² + k
  • Vertex is at (h, k). The sign flips: y = (x βˆ’ 3)Β² has vertex at x = 3, not βˆ’3.
  • a > 0: opens up (minimum). a < 0: opens down (maximum).
SAT Example
y = 2(x βˆ’ 4)Β² + 1: vertex at (4, 1), opens upward, minimum value is 1.
Quadratics β€” Shortcut

Key Factoring Identities

(a+b)Β² = aΒ²+2ab+bΒ²
(aβˆ’b)Β² = aΒ²βˆ’2ab+bΒ²
(a+b)(aβˆ’b) = aΒ²βˆ’bΒ²
  • The difference of squares aΒ²βˆ’bΒ² shows up constantly. Recognise it instantly.
SAT Example
Factor xΒ²βˆ’16: difference of squares β†’ (x+4)(xβˆ’4). Factor xΒ²+6x+9: perfect square β†’ (x+3)Β².
07 Coordinate Geometry
Coordinate Geometry

Distance Between Two Points

d = √[(xβ‚‚βˆ’x₁)Β² + (yβ‚‚βˆ’y₁)Β²]
  • This is just the Pythagorean theorem applied to coordinates β€” the distance is the hypotenuse.
SAT Example
Distance from (1,2) to (4,6): √[(3)²+(4)²] = √25 = 5.
Coordinate Geometry

Midpoint Formula

M = ((x₁+xβ‚‚)/2, (y₁+yβ‚‚)/2)
  • Average the x-coordinates, average the y-coordinates.
SAT Example
Midpoint of (2,4) and (8,10): ((2+8)/2, (4+10)/2) = (5, 7).
08 Statistics & Probability
Statistics β€” Core

Mean, Median, Mode, Range

Mean = Ξ£x / n
  • Mean: sum of all values Γ· count of values.
  • Median: middle value when sorted. For even count: average of two middle values.
  • Mode: value that appears most often.
  • Range: max βˆ’ min.
  • SAT tip: to find a missing value given a target mean, use total = mean Γ— n then subtract the known values.
SAT Example
Data: {3, 5, 7, 9, 11}. Mean = 35/5 = 7. Median = 7. Range = 11βˆ’3 = 8. If the mean needs to be 8, the sum needed = 40, so missing value = 40βˆ’35 = 5.
Probability

Basic Probability

P(event) = favourable outcomes / total outcomes
  • P(A and B) = P(A) Γ— P(B) if events are independent.
  • P(A or B) = P(A) + P(B) βˆ’ P(A and B).
  • P(not A) = 1 βˆ’ P(A).
SAT Example
Bag has 4 red, 6 blue marbles. P(red) = 4/10 = 0.4. P(not red) = 1 βˆ’ 0.4 = 0.6.
09 Angles & Polygons
Angles β€” Core

Angle Relationships

Vertical angles equal. Supplementary = 180Β°. Full turn = 360Β°.
  • Vertical angles (X-shaped): always equal.
  • Linear pair: two angles on a straight line add to 180Β°.
  • Transversal cutting parallel lines: alternate interior angles equal, corresponding angles equal, co-interior angles sum to 180Β°.
Ξ± Ξ± alternate interior angles equal
SAT Example
Two angles form a linear pair: one is 65Β°. The other = 180 βˆ’ 65 = 115Β°.
Polygons

Triangle & Polygon Angle Sums

Triangle: 180Β°  |  Polygon: (nβˆ’2) Γ— 180Β°
  • Sum of interior angles of any triangle is always 180Β°.
  • For any polygon with n sides: (nβˆ’2) Γ— 180. Quadrilateral = 360Β°, pentagon = 540Β°, hexagon = 720Β°.
  • Exterior angle of a triangle = sum of the two non-adjacent interior angles.
  • Sum of exterior angles of any polygon = always 360Β°.
SAT Example
Hexagon interior angle sum: (6βˆ’2) Γ— 180 = 720Β°. Regular hexagon: each angle = 720/6 = 120Β°.
10 Trigonometry & Circles
Trigonometry β€” High Frequency

SOHCAHTOA

sin = O/H   cos = A/H   tan = O/A
  • O = opposite, A = adjacent, H = hypotenuse β€” relative to the angle ΞΈ.
  • Co-function identity: sin(ΞΈ) = cos(90Β°βˆ’ΞΈ). These come up on the SAT specifically.
  • Memorise sin/cos/tan for 30Β°, 45Β°, 60Β° if the question doesn't give you a calculator.
ΞΈ A (adjacent) O H (hyp)
SAT Example
In a right triangle, opposite = 3, hypotenuse = 5. sin(ΞΈ) = 3/5. Also: cos(90Β°βˆ’ΞΈ) = 3/5.
Circles

Arc Length & Sector Area

Arc = (ΞΈ/360) Γ— 2Ο€r  |  Sector = (ΞΈ/360) Γ— Ο€rΒ²
  • ΞΈ is the central angle in degrees. The fraction ΞΈ/360 is just the proportion of the full circle.
  • Central angle = arc it subtends. Inscribed angle = half the central angle for the same arc.
SAT Example
Circle radius 6, central angle 90Β°. Arc = (90/360) Γ— 2Ο€(6) = ΒΌ Γ— 12Ο€ = 3Ο€. Sector area = ΒΌ Γ— Ο€(36) = 9Ο€.
Circles

Equation of a Circle

(x βˆ’ h)Β² + (y βˆ’ k)Β² = rΒ²
  • Centre is at (h, k). Watch the sign flip: (xβˆ’3)Β² means centre x = 3, not βˆ’3.
  • If the equation isn't in this form, complete the square on both x and y terms.
SAT Example
(x+2)Β² + (yβˆ’5)Β² = 49: centre = (βˆ’2, 5), radius = 7.
Trigonometry

Radians & Degrees

Ο€ radians = 180Β°  |  Radians = Degrees Γ— Ο€/180
  • To convert degrees β†’ radians: multiply by Ο€/180. Radians β†’ degrees: multiply by 180/Ο€.
  • Key conversions: 30Β° = Ο€/6, 45Β° = Ο€/4, 60Β° = Ο€/3, 90Β° = Ο€/2, 180Β° = Ο€.
SAT Example
Convert 270Β° to radians: 270 Γ— Ο€/180 = 3Ο€/2.
11 Function Transformations
Transformations β€” High Frequency

Graph Shifts

f(x) + k  |  f(x βˆ’ h)  |  βˆ’f(x)  |  f(βˆ’x)
  • f(x) + k: shift up k units. f(x) βˆ’ k: shift down.
  • f(x βˆ’ h): shift right h units. f(x + h): shift left. (Counter-intuitive β€” the sign flips.)
  • βˆ’f(x): reflect over x-axis. f(βˆ’x): reflect over y-axis.
  • aΒ·f(x): vertical stretch (a > 1) or compression (0 < a < 1).
f(x)+k ↑ f(xβˆ’h) β†’ original
SAT Example
f(x) = xΒ². g(x) = (xβˆ’3)Β² + 2 shifts the parabola right 3 and up 2. New vertex: (3, 2).

Common SAT Math Mistakes

βœ•

Dividing by the wrong base

Percent change always divides by the original (old) value β€” not the new one. Mixing these up is the most common percent error.

βœ•

Horizontal shift sign flip

f(x βˆ’ 3) shifts the graph right, not left. The sign inside the bracket is opposite to the direction of the shift.

βœ•

Forgetting the Β± in the quadratic formula

The Β± gives two solutions. Missing the negative case means losing one root β€” and the question may specifically ask for the negative one.

βœ•

Confusing radius and diameter

Area and circumference formulas use radius. If a problem gives diameter, halve it first β€” or use Ο€d for circumference only.

βœ•

Chaining percentages additively

Two 10% increases is not a 20% increase. It's 1.1 Γ— 1.1 = 1.21 β€” a 21% increase. Always use the multiplier method.

βœ•

Height in triangle formulas

Height must always be perpendicular to the base. In an obtuse triangle, the height may fall outside the triangle β€” don't use a side length as the height.

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