Digital SAT Math · 51 formulas
DSAT Math Formula Vault
Every formula you need for the Digital SAT Math section, grouped by topic. Each card shows what the formula means and where it is used; open “Deep dive” for common mistakes, shortcuts and worked examples.
By Engin Savaş · MATHIGH · mathematics educator with 20+ years of AP and IB teaching
01Algebra
Linear equations and lines
1.1 Lines
Slope-intercept form
The line with slope and y-intercept .
SAT useModeling, rates and predictions.
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Meaning
is the change in when increases by 1; is the value of when .
Context words
“Per”, “each” and “every” signal the slope. “Initially”, “at time 0” and “fixed fee” signal the intercept.
Example
$5 per hour plus a $10 base fee gives .
Graph
: the line rises. : the line falls. : the line is horizontal.
Slope from two points
The change in per unit change in .
SAT useTables, graphs and rates of change.
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Common mistake
Mixing the order of the points. Subtract in the same order on top and bottom.
Special cases
A horizontal line has slope 0. A vertical line has an undefined slope (the denominator is 0).
Tip
When is time, the slope is the rate at which the quantity changes.
Point-slope form
The line through with slope .
SAT useWriting a line from one point and a rate.
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Meaning
and are fixed numbers: the coordinates of a known point on the line.
Example
Slope 3 through : , which is .
Tip
Use it when a question gives “at the value is ” together with a rate.
Standard form
A line written with both variables on one side.
SAT useBudgets and combinations, such as tickets of two prices.
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Slope
Rearranging gives slope and y-intercept .
Intercepts
Set for the y-intercept and for the x-intercept.
Example
Adult tickets cost $8 and child tickets $5, with $200 collected: .
1.2 Systems of two linear equations
Number of solutions
Compare the slopes and intercepts of the two lines.
SAT use“No solution” and “infinitely many solutions” questions.
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One solution
Different slopes: the lines cross at one point.
No solution
Same slope, different intercepts: the lines are parallel.
Infinitely many
One equation is a multiple of the other: the lines are the same.
Example
and have infinitely many solutions when and no solution for any other .
02Advanced Math
Quadratics
2.1 Forms of a quadratic
Standard form
The general form of a quadratic.
SAT useThe y-intercept, the direction of opening and the discriminant.
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Reading it
: opens upward (minimum). : opens downward (maximum). is the y-intercept.
Common mistake
Reading the vertex from alone. The vertex is at , which depends on both and .
Tip
For roots, factor or use the quadratic formula; for the number of roots, use the discriminant.
Vertex form
The vertex is ; controls direction and width.
SAT useMaximum and minimum questions.
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Reading it
: the vertex is a minimum. : it is a maximum. A larger gives a narrower parabola.
Sign check
means ; means .
Tip
Maximum height, maximum profit and minimum cost are vertex questions: use vertex form or .
Factored form
The zeros are and .
SAT useRoots, x-intercepts and solving equations.
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Axis of symmetry
, halfway between the roots.
Common mistake
Sign errors: the factor gives the root .
Example
has roots 2 and 8, axis and vertex .
Quadratic formula
The solutions of .
SAT useWhen factoring is not quick or the roots are irrational.
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Tip
Work out the discriminant first: it tells you how many solutions to expect.
Common mistake
Dividing only part of the numerator by , or dropping the .
Shortcut
The sum of the roots is and the product is .
2.2 Graph behaviour
Axis of symmetry
The vertical line through the vertex.
SAT useFinding the vertex from standard form.
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Meaning
The parabola is symmetric about this line, and the vertex lies on it.
Tip
In a table, two equal y-values sit symmetrically: the axis is halfway between their x-values.
Vertex value
The maximum or minimum value of the function.
SAT useOptimisation: greatest height, least cost.
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Meaning
A minimum when , a maximum when .
Example
: and the maximum height is .
Discriminant
Decides how many real solutions there are.
SAT use“How many solutions?” and “for what value of …?”
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Cases
: two x-intercepts. : one (the vertex touches the axis). : none.
Example
has exactly one solution when , so .
03Advanced Math
Exponential functions
3.1 Growth and decay models
General exponential
Start at and multiply by each step.
SAT useRecognising and building exponential models.
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Reading it
: growth. : decay. is the value when .
Linear or exponential?
Linear models add the same amount each step; exponential models multiply by the same factor.
Common mistake
Confusing with the rate: is the multiplier, not the percentage.
Percentage growth
Growth by rate per period, with as a decimal.
SAT useInterest, population, compound growth.
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Example
12% growth per year: and .
Common mistake
Adding the percentage each year instead of compounding it.
Tip
“After 3 years” means an exponent of 3, not multiplying by 3.
Percentage decay
Decay by rate per period, with as a decimal.
SAT useDepreciation and value loss.
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Example
A 20% loss per year gives the multiplier 0.80.
Repeated losses
Two 20% losses: , a 36% loss, not 40%.
Graph
The graph decreases towards 0 but never reaches it.
3.2 Doubling and half-life
Doubling every d units
The quantity doubles every units of time.
SAT use“Doubles every 5 years”.
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Example
A colony of 300 bacteria doubles every 4 hours: .
Check
When , the exponent is 1 and .
Half-life h
The quantity halves every units of time.
SAT useRadioactive decay, medicine in the body.
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Example
80 mg with a half-life of 6 hours: ; after 12 hours, 20 mg remain.
Check
When , the exponent is 1 and .
Reaching a target
Find when an exponential model reaches a given value.
SAT use“After how many years will the value exceed …?”
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On the SAT
Graph and in Desmos and read the intersection, or test the answer choices. Logarithms are not required.
Beyond the SAT
Algebraically, .
3.3 Percentages as multipliers
Increase by r%
Multiply by 1 plus the rate.
SAT usePrice rises, growth.
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Example
+20% → × 1.20; +7.5% → × 1.075.
Why
The new amount is the original plus of the original.
Decrease by r%
Multiply by 1 minus the rate.
SAT useDiscounts, depreciation.
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Example
−30% → × 0.70.
Common mistake
Using 0.30 for “decreases by 30%”: 0.30 is what is lost, 0.70 is what remains.
Repeated change
The same percentage change applied times.
SAT useCompound growth over several periods.
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Rule
Repeated changes multiply, so they become an exponent.
Different changes
+20% then −10%: , an 8% increase overall.
04Problem-Solving and Data
Problem-solving and data
4.1 Percentages and percent change
Percent of a whole
What share a part is of the whole.
SAT useData tables, shares, distributions.
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Key question
Ask “out of what?”: many questions change the whole.
Tip
For percent increase, the whole is the original value. For percent of total, the whole is the total.
Percent change
The relative increase or decrease.
SAT usePrices, populations, changes in data.
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Common mistake
Dividing by the new value instead of the old one.
Reverse changes
If A → B is +30%, then B → A is , not −30%.
Successive discounts
20% then 30% off: , a 44% decrease in total.
4.2 Ratios, proportions and scale
Ratio
A comparison of two quantities by division.
SAT useMixtures, scale models, part-to-part comparisons.
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Part of the total
If , then is of the total.
Example
and give and .
Proportion
Two equal ratios.
SAT useMissing-value and unit-rate questions.
Deep dive
Common mistake
Writing the two ratios in different orders.
Tip
Keep units aligned: boys/girls = boys/girls.
Scale factor
How many times larger or smaller something becomes.
SAT useSimilar figures and scale drawings.
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Scaling rule
Lengths × , areas × , volumes × .
Example
Similar triangles with sides in the ratio 2 : 3: if the smaller area is 20, the larger is .
4.3 Rates
Speed and rates
How much per one unit of time.
SAT useTravel, efficiency, slope as a rate.
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Example
150 miles in 3 hours: mph.
Common mistake
Not converting units first, such as minutes to hours.
Connection
In a linear model, the slope is the unit rate.
Unit cost
The cost of one unit.
SAT use“Which is the better buy?”
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Rule
The lower unit cost is the better buy.
Common mistake
Comparing total prices for different quantities.
Combined work
Work rates add; times do not.
SAT useTwo machines or people working together.
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Method
Convert “hours per job” into “jobs per hour”, then add.
Example
One pump fills a tank in 3 h and another in 6 h: tank per hour, so together they take 2 h.
4.4 Means
Mean
The sum of the values divided by how many there are.
SAT useMissing values and data interpretation.
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Tip
If one value changes by , the mean changes by .
Adding a value
If the mean rises after a value is added, the new value is above the old mean.
Weighted mean
A mean in which values count different numbers of times.
SAT useCombining groups of different sizes.
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Common mistake
Averaging the group means; that only works when the groups are the same size.
Example
20 students averaging 80 and 30 averaging 70: .
Total from a mean
Rebuild a total from a mean and a count.
SAT useFinding a missing score.
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Method
Find the total needed for the target mean, subtract the known values; what is left is the missing value.
Example
Four scores average 85. What fifth score gives a mean of 87? .
05Geometry and Trigonometry
Geometry and trigonometry
5.1 Area, perimeter and circles
Rectangle
Area of a rectangle.
SAT useComposite shapes and floor plans.
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Scaling
If every length is multiplied by , the perimeter is multiplied by and the area by .
Triangle
Area of a triangle.
SAT useComposite shapes and heights.
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Key idea
The height is perpendicular to the base.
Common mistake
Using a slanted side as the height.
Circle area
Area of a circle of radius .
SAT useCircular regions and rings.
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Scaling
Doubling multiplies the area by 4.
Ring
The area between circles of radii and is .
Circumference
The distance around a circle.
SAT useWheels, rotations, perimeters.
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Example
A wheel of diameter 0.7 m travels m per rotation.
Sector area
The fraction of the circle cut out by the angle .
SAT useShaded regions, pie charts.
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Radians
With in radians, .
Example
A 90° sector of a circle of radius 6: .
Arc length
The fraction of the circumference cut out by the angle .
SAT useArcs and rotations.
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Radians
With in radians, .
Conversion
radians.
5.2 Coordinate geometry
Distance
The distance between two points.
SAT useLengths on the coordinate plane.
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Key idea
It is Pythagoras: the horizontal and vertical differences are the legs.
Example
From to : .
Midpoint
The point halfway along a segment.
SAT useSymmetry and the centre of a circle from a diameter.
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Tip
Average the x-coordinates and average the y-coordinates.
Parallel and perpendicular
Parallel lines share a slope; perpendicular slopes are negative reciprocals.
SAT useWriting parallel or perpendicular lines.
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Example
A line perpendicular to has slope .
Connection
Parallel lines never meet, so a system of two parallel lines has no solution.
Circle equation
A circle with centre and radius .
SAT useReading centre and radius; completing the square.
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Sign check
means .
Example
becomes : centre , radius 5.
5.3 Triangles
Pythagorean theorem
The side relationship in a right triangle.
SAT useMissing sides and hidden right angles.
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Key idea
It works only in right triangles; is the hypotenuse.
Common triples
3-4-5, 5-12-13, 8-15-17 and their multiples.
45-45-90 triangle
An isosceles right triangle.
SAT useSquares and their diagonals.
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Tip
The diagonal of a square of side is .
Recognise it
A right triangle with two equal legs is a 45-45-90 triangle.
30-60-90 triangle
Half of an equilateral triangle.
SAT useHeights of equilateral triangles.
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Key idea
The hypotenuse is twice the shorter leg; the longer leg is opposite 60°.
Tip
A in the answer choices often points to a 30-60-90 triangle.
5.4 Trigonometry
SOH CAH TOA
The trigonometric ratios in a right triangle.
SAT useHeights, shadows, angles of elevation and depression.
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Labelling
Opposite and adjacent depend on which angle you use.
Angles
Elevation is measured up from the horizontal; depression is measured down from the horizontal.
Complement rule
Sine and cosine of complementary angles are equal.
SAT useOne of the most frequent SAT trigonometry questions.
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Why
The side opposite one acute angle is adjacent to the other, and the two acute angles add to 90°.
Example
If , then .
Pythagorean identity
Links sine and cosine of the same angle.
SAT useFinding one ratio from another.
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Example
If for an acute angle, then .
5.5 Volume
Cylinder
Base area times height.
SAT useTanks, cans, pipes.
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Scaling
Halving and tripling multiplies the volume by .
Sphere
Volume of a ball of radius .
SAT useBalls, domes, scaling.
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Scaling
Doubling the radius multiplies the volume by 8.
Cone and pyramid
One third of the matching cylinder or prism.
SAT useFunnels, roofs, comparisons.
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Tip
All of these formulas are on the reference sheet in Bluebook.
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