Applications and Interpretation SL focuses on modelling with functions, statistics, finance and using a graphic display calculator. The IB formula booklet gives you the formulas with no explanation. Here are all 51 that apply to AI SL, with what each one means and when to use it.
This is an annotated companion to the booklet you are given in the exam, not a copy of it. Formulas are grouped by topic: Prior learning, Number & Algebra, Functions, Geometry & Trigonometry, Statistics & Probability, Calculus.
Base times perpendicular height.
When: Any parallelogram.
Half of base times perpendicular height.
When: When you know a base and its height.
Average of the two parallel sides times the height between them.
When: Four-sided shapes with one pair of parallel sides.
π times the radius squared.
When: Circles and disc-shaped regions.
The distance around a circle.
When: Perimeter of a circle.
Length × width × height.
When: Rectangular boxes.
Circle area times height.
When: Cylinders / tubes.
Cross-section area times length.
When: Any solid with a constant cross-section.
Unroll the side into a rectangle: circumference × height.
When: Surface area of the curved part.
Pythagoras on the horizontal and vertical gaps.
When: Length of a segment from coordinates.
Average the x-coordinates and the y-coordinates.
When: The point halfway between two points.
Start at the first term and add the common difference once per step.
When: Sequences that go up/down by a fixed amount.
Use when you know the first term and the common difference.
When: Totalling an arithmetic sequence from \(u_1\) and \(d\).
Use when you know the first and last term.
When: Totalling an arithmetic sequence from \(u_1\) and \(u_n\).
Multiply the first term by the ratio once per step.
When: Growth/decay by a fixed factor.
Shortcut for adding terms that keep multiplying by r.
When: Repeated percentage growth (e.g. savings).
PV = start; r% = yearly rate; k = compounds per year; n = years.
When: Investments/loans compounding more than once a year.
How far an approximate value is from the exact one, as a %.
When: Rounded/measured vs true value.
Logs undo exponentials; use a log when the unknown is a power.
When: Solving for an exponent.
The vertical line through the vertex - halfway between the roots.
When: Max/min of a quadratic.
Three equivalent forms: gradient–intercept, general, and point–gradient.
When: Lines, gradients and intercepts.
Rise over run between two points.
When: Finding the slope from two points.
Grows with the cube of the radius.
When: Balls, domes.
Exactly four circle-areas.
When: Outer area of a sphere.
A third of the cylinder with the same base and height.
When: Cones.
Uses the slant height l, not the vertical height.
When: Surface of the sloping part.
A third of base area times height.
When: Pyramids.
The three ratios linking an angle to the sides of a right triangle.
When: Right-angled triangles.
Pairs each side with its opposite angle.
When: A side + its opposite angle are known.
Pythagoras plus an angle-correction term.
When: Two sides + included angle, or all three sides.
Half the product of two sides times the sine of the angle between.
When: No perpendicular height available.
The fraction θ/360 of the full circumference.
When: AI - angles in degrees.
The fraction θ/360 of the full circle area.
When: AI - angles in degrees.
Pythagoras in three dimensions.
When: Length of a segment in 3-D space.
Average each coordinate of the two endpoints.
When: Midpoint in 3-D space.
Each value weighted by how often it occurs.
When: Grouped or repeated data.
Favourable outcomes over total outcomes.
When: Equally-likely outcomes.
Add the chances, subtract the overlap.
When: P(A or B).
Chance of A once B is known - sample space shrinks to B.
When: 'Given that…' problems.
Multiply when one event doesn't affect the other.
When: Both happen; also the test for independence.
Long-run average: outcomes weighted by probability.
When: Fair games, average payoff.
n independent trials, success chance p; on average np successes.
When: Fixed number of yes/no trials.
Line of best fit; r measures strength/direction of linear association.
When: Bivariate data, prediction.
Compares observed with expected frequencies.
When: Tests of independence / goodness-of-fit (AI).
The spread of the middle 50% of the data.
When: Spread; outlier boundaries.
An event and its non-occurrence have probabilities that sum to 1.
When: Finding \(P(\text{not } A)\).
When two events cannot both happen, just add their probabilities.
When: Events with no overlap.
Bring the power down, reduce it by one.
When: Differentiating powers of x.
Reverse of differentiating: raise the power, divide by it.
When: Integrating powers of x (don't forget +C).
A definite integral sums thin strips into a signed area.
When: Area, and (with velocity) distance.
Estimate area with trapezia of width h.
When: Approximating an integral (AI).
Knowing a formula is not the same as using it under exam pressure. Practise every AI SL topic with auto-marked questions and full worked solutions, and see the GDC guides for the calculator steps that go with them.