Should I memorize formulas for the OAR?

Yes. Even where a formula might be given, recalling it instantly saves time, and you need to know which one a question calls for. Practice using them on paper, since calculators aren’t allowed.

Which formulas matter most for the OAR?

For Math Skills: distance = rate × time, percents and percent change, ratios, area and perimeter, the Pythagorean theorem, mean and median. For Mechanical Comprehension: the lever balance, pulley and ramp mechanical advantage, gear speed, pressure = force ÷ area and Ohm’s law.

How should I memorize these formulas?

Use each one, don’t just read it. Work one or two problems per formula, then mix them in timed practice. The OAR Prep app has flashcards for most of these and 2,604 practice questions that use them.

Free tool

OAR formula sheet

The math and mechanical formulas behind most OAR questions, grouped by topic, each with a quick example. Print it or keep it open while you practice.

47 formulas in 8 groups. Print-friendly: use your browser’s print command.

Math Skills

Arithmetic, fractions and percents

Order of operations (PEMDAS): Parentheses → Exponents → × and ÷ (left to right) → + and − (left to right); 3 + 4 × 2 = 3 + 8 = 11
Multiplying fractions: a/b × c/d = (a × c) / (b × d); 2/3 × 3/4 = 6/12 = 1/2
Dividing fractions: a/b ÷ c/d = a/b × d/c; 1/2 ÷ 3/4 = 1/2 × 4/3 = 2/3
Percent of a number: part = percent (as a decimal) × whole; 25% of 200 = 0.25 × 200 = 50
Percent change: (new − old) ÷ old × 100; $80 → $60: (60 − 80) ÷ 80 = −25%
Ratio and proportion: a/b = c/d ⇒ a × d = b × c; 3/4 = x/20 ⇒ 4x = 60, x = 15
Exponents: aᵐ × aⁿ = aᵐ⁺ⁿ · aᵐ ÷ aⁿ = aᵐ⁻ⁿ · a⁰ = 1; 2³ × 2⁴ ÷ 2² = 2⁵ = 32
Square root: √x × √x = x; √49 = 7 because 7 × 7 = 49

Math Skills

Algebra and word problems

Distance, rate, time: d = r × t · r = d ÷ t · t = d ÷ r; 150 miles in 3 hours = 50 mph
Average speed (round trip): total distance ÷ total time (not the mean of the speeds); Equal legs at 40 and 60 mph: 2 × 40 × 60 ÷ 100 = 48 mph
Combined work rate: 1/T = 1/A + 1/B; 6 h and 12 h pumps: 1/6 + 1/12 = 1/4, so 4 hours together
Simple interest: I = P × r × t; $2,000 at 6% for 3 years = $360
Consecutive integers: n, n + 1, n + 2 …; Two that sum to 47: 2n + 1 = 47, so 23 and 24
Slope of a line: m = (y₂ − y₁) ÷ (x₂ − x₁); (1, 2) to (3, 8): 6 ÷ 2 = 3

Math Skills

Geometry

Rectangle: Area = length × width · Perimeter = 2(l + w); 8 ft × 5 ft = 40 ft²
Triangle: Area = ½ × base × height; base 10, height 6: area 30
Angles: Triangle: 180° · Supplementary: 180° · Complementary: 90°; Ratio 2:6:10 → 180 ÷ 18 = 10°, smallest angle 20°
Pythagorean theorem: a² + b² = c² (c is the hypotenuse); 3² + 4² = 25, so c = 5
Circle: Area = πr² · Circumference = 2πr = πd; d = 10 in: C = 3.14 × 10 = 31.4 in
Rectangular box: Volume = length × width × height; 2 × 3 × 4 = 24 cubic units
Cylinder: Volume = πr² × h; r = 3 ft, h = 10 ft: 3.14 × 9 × 10 ≈ 283 ft³

Math Skills

Averages and probability

Mean: sum of the values ÷ number of values; (4 + 8 + 9) ÷ 3 = 7
Median: the middle value in order (average the two middle values if the count is even); 68, 72, 76, 84, 90 → 76
Mode: the value that appears most often; 42, 48, 50, 50, 60 → 50
Probability: favorable outcomes ÷ possible outcomes (0 to 1); Two reds from 4 red + 6 blue: 4/10 × 3/9 = 2/15

Mechanical Comprehension

Simple machines

Mechanical advantage: MA = output force ÷ input force = effort distance ÷ load distance; A machine with MA 4 lifts 400 lb with 100 lb
Levercalculator: effort × effort arm = load × load arm; 300 lb at 2 ft balanced 4 ft away by 150 lb
Pulley systemcalculator: MA = number of rope segments supporting the load; 4 segments: 400 lb needs 100 lb of pull
Inclined planecalculator: MA = length ÷ height · effort = load × height ÷ length; 30 lb up a 15 ft ramp rising 3 ft: 6 lb
Wheel and axlecalculator: MA = wheel radius ÷ axle radius; 12 in handle on a 2 in axle: MA 6
Gearscalculator: teeth × rpm is equal across meshed gears; each mesh reverses direction; 20 teeth at 60 rpm drives 40 teeth at 30 rpm

Mechanical Comprehension

Force, motion and energy

Newton’s second law: F = m × a; More force, more acceleration; more mass, less
Work: W = force × distance (in the direction of the force); 50 lb pushed 10 ft = 500 ft·lb
Power: P = work ÷ time; The same work in half the time takes twice the power
Torque: τ = force × distance from the pivot; A longer wrench gives more torque for the same pull
Momentum: p = m × v; 20 kg at 20 m/s = 400 kg·m/s
Kinetic energy: KE = ½ × m × v²; Double the speed, four times the energy
Potential energy: PE = m × g × h; Lift it twice as high, twice the energy

Mechanical Comprehension

Pressure and fluids

Pressure: P = force ÷ area; 600 lb on 4 ft² = 150 lb/ft²
Hydraulic press (Pascal)calculator: F₁ ÷ A₁ = F₂ ÷ A₂; 50 lb on 2 in² lifts 500 lb on 20 in²
Density: ρ = mass ÷ volume; Less dense than the fluid: it floats
Buoyancy (Archimedes): buoyant force = weight of the fluid displaced; 100 lb object displacing 30 lb of water weighs 70 lb underwater
Flow through a pipe: A₁ × v₁ = A₂ × v₂; Halve the area and the water moves twice as fast

Mechanical Comprehension

Electricity

Ohm’s lawcalculator: V = I × R; 12 V across 4 Ω drives 3 A
Electrical power: P = V × I; 12 V × 3 A = 36 W
Resistors in seriescalculator: R = R₁ + R₂ + …; 2 + 2 + 2 = 6 Ω
Resistors in parallelcalculator: 1/R = 1/R₁ + 1/R₂ + …; Two 4 Ω in parallel = 2 Ω

How to use this sheet

The formulas cover the two OAR subtests that involve calculation: Math Skills and Mechanical Comprehension. Calculators aren’t allowed, so each example uses numbers you can work on paper. Mechanical formulas describe ideal machines with no friction, which is how practice questions treat them unless they say otherwise.

To put them to work, try the math andmechanical comprehension practice tests, or theno-calculator math drill.

Test yourself: an OAR-style question

A candidate scores 80 on three exams and 90 on two exams. What is the average across all five exams?

Show answer

C. 84

Weight each score by how many times it occurs: (3 × 80 + 2 × 90) ÷ 5 = 420 ÷ 5 = 84. Averaging 80 and 90 to get 85 ignores the weights.

Questions like this come up on the Math Skills section of the OAR. The 2,604 questions in the app each come with a written explanation.

FAQ

Should I memorize formulas for the OAR?: Yes. Even where a formula might be given, recalling it instantly saves time, and you need to know which one a question calls for. Practice using them on paper, since calculators aren’t allowed.
Which formulas matter most for the OAR?: For Math Skills: distance = rate × time, percents and percent change, ratios, area and perimeter, the Pythagorean theorem, mean and median. For Mechanical Comprehension: the lever balance, pulley and ramp mechanical advantage, gear speed, pressure = force ÷ area and Ohm’s law.
How should I memorize these formulas?: Use each one, don’t just read it. Work one or two problems per formula, then mix them in timed practice. The OAR Prep app has flashcards for most of these and 2,604 practice questions that use them.

Sources

Checked against these sources on October 24, 2026. Mechanical formulas assume ideal machines (no friction). For OAR preparation only; not affiliated with or endorsed by the U.S. Navy.