Here is a complete, organized list of ALL important UPCAT Math
formulas you'll need — simplified and grouped by topic, perfect for
quick review and memorization:
ARITHMETIC & NUMBER THEORY
➊ Divisibility Rules (Quick Mental Checks)
2: Last digit even
3: Sum of digits divisible by 3
4: Last 2 digits divisible by 4
5: Ends in 0 or 5
6: Divisible by 2 and 3
8: Last 3 digits divisible by 8
9: Sum of digits divisible by 9
10: Ends in 0
11: Alternating sum and difference divisible by 11
➋ LCM & GCF
GCF: Product of lowest powers of common primes
LCM: Product of highest powers of all primes
➌ Percent, Ratio, and Proportion
% to decimal: ÷100
% increase: new - old / old × 100
% decrease: old - new / old × 100
Partitive proportion: Total parts
📐 GEOMETRY
Divide accordingly
➊ Angles
Triangle: 180°
Quadrilateral: 360°
Straight line: 180°
Circle: 360°
Exterior angle of polygon: 360° ÷ n
Interior angle of polygon: [(n – 2) × 180°] ÷ n
➋ Triangle Area
Base × Height: ½ × b × h
Heron’s Formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
➌ Circle
Area: πr²
Circumference: 2πr
Arc Length: (θ / 360) × 2πr
Sector Area: (θ / 360) × πr²
➍ Rectangle, Square, Parallelogram
Rectangle Area: l × w
Square Area: s²
Parallelogram: b × h
Trapezoid: ½ × (b₁ + b₂) × h
🧮 ALGEBRA
➊ Special Products
(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
(a + b)(a – b) = a² – b²
➋ Factoring Patterns
Common Monomial
Difference of Squares: a² – b² = (a + b)(a – b)
Perfect Square Trinomial
Trinomial: ax² + bx + c
🧠 WORD PROBLEMS
➊ Age / Work / Motion
Work: 1 / time₁ + 1 / time₂ = 1 / total time
Motion: Distance = Rate × Time
Opposite Direction: Add speeds
Same Direction: Subtract speeds
➋ Mixture / Percent
Total = (% × amount) + (% × amount)
Use weighted average or alligation if needed.
📊 STATISTICS & PROBABILITY
➊ Mean, Median, Mode
Mean: Sum ÷ Count
Median: Middle value
Mode: Most frequent
➋ Probability
P = favorable outcomes / total outcomes
AND: Multiply probabilities
OR (mutually exclusive): Add probabilities
🧮 EXPONENTS & RADICALS
➊ Laws of Exponents
a^m × a^n = a^(m+n)
a^m ÷ a^n = a^(m-n)
(a^m)^n = a^(mn)
a^0 = 1
a^(-n) = 1 / a^n
➋ Radicals
√(a²) = |a|
√(ab) = √a × √b
√(a/b) = √a ÷ √b
Rationalize denominators with radicals.
📈 LINEAR EQUATIONS & SYSTEMS
Slope formula: (y₂ – y₁) / (x₂ – x₁)
Point-slope form: y – y₁ = m(x – x₁)
Slope-intercept: y = mx + b
Standard form: Ax + By = C
📐 MENSURATION & SOLID GEOMETRY
➊ Volume
Cube: s³
Rectangular Prism: l × w × h
Cylinder: πr²h
Cone: (1/3)πr²h
Sphere: (4/3)πr³
➋ Surface Area
Cube: 6s²
Rectangular Prism: 2(lw + lh + wh)
Cylinder: 2πr² + 2πrh
🔤 SEQUENCES & SERIES
➊ Arithmetic Sequence
nth term: aₙ = a₁ + (n – 1)d
Sum: Sₙ = n/2 [2a + (n – 1)d]
➋ Geometric Sequence
nth term: aₙ = a₁ × rⁿ⁻¹
Sum (finite): Sₙ = a(1 – rⁿ) / (1 – r)
Sum (infinite): S = a / (1 – r) if |r| < 1
✳️QUADRATIC EQUATIONS
Standard form: ax² + bx + c = 0
Quadratic Formula: x = [-b ± √(b² – 4ac)] / 2a
Discriminant: D = b² – 4ac
D>0
2 real roots
D=0
1 real root
D<0
no real root