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Core Formula Reference

Essential AMC8 formulas covering arithmetic, algebra, geometry, number theory, combinatorics and probability for quick pre-exam review.

Quick Reference

AMC8 Essential Formulas

Organized by topic for quick review before the exam

Name	Formula	Notes
Sum of Arithmetic Sequence	S = n(a₁ + aₙ) / 2	n = number of terms, a₁ = first term, aₙ = last term
nth Term of Arithmetic Sequence	aₙ = a₁ + (n-1)d	d = common difference
Sum of Geometric Sequence	S = a₁(1 - rⁿ) / (1 - r)	r ≠ 1, r = common ratio
Mean (Average)	x̄ = Σxᵢ / n	sum of all data divided by count
Percentage Change	new = original × (1 ± p%)	increase: +, decrease: -
Quadratic Formula	x = (-b ± √(b²-4ac)) / 2a	discriminant Δ = b² - 4ac
Absolute Value Inequality	\|x\| < a → -a < x < a	a > 0
Factoring	a²-b² = (a+b)(a-b)	difference of squares
Perfect Square	(a±b)² = a² ± 2ab + b²	note the middle term coefficient 2

Name	Formula	Notes
Triangle Area	S = ½ × base × height	also S = ½ab·sinC
Pythagorean Theorem	a² + b² = c²	c is the hypotenuse
Circle Area	S = πr²	r = radius
Circumference	C = 2πr	r = radius
Trapezoid Area	S = ½(a+b) × h	a, b = bases, h = height
Rhombus Area	S = ½d₁ × d₂	d₁, d₂ = diagonals
Rectangular Prism Volume	V = l × w × h	length × width × height
Cylinder Volume	V = πr²h	r = base radius, h = height
Sphere Volume	V = 4/3 × πr³	r = radius
Midpoint Formula	M = ((x₁+x₂)/2, (y₁+y₂)/2)	midpoint of two points
Distance Formula	d = √((x₂-x₁)² + (y₂-y₁)²)	distance between two points on coordinate plane

Name	Formula	Notes
Number of Divisors	n = p₁^a₁ × p₂^a₂ → (a₁+1)(a₂+1)	multiply (exponent+1) for each prime factor
GCD × LCM	GCD(a,b) × LCM(a,b) = a × b	product of GCD and LCM of two numbers
Divisibility Rule (3)	digit sum divisible by 3	e.g., 123 → 1+2+3=6 ✓
Divisibility Rule (9)	digit sum divisible by 9	e.g., 729 → 7+2+9=18 ✓
Divisibility Rule (11)	alternating digit sum divisible by 11	e.g., 121 → 1-2+1=0 ✓
Legendre's Formula	power of p in n! = Σ⌊n/p^k⌋	count prime factors in factorial

Name	Formula	Notes
Permutation	P(n,r) = n! / (n-r)!	choose r from n, order matters
Combination	C(n,r) = n! / (r!(n-r)!)	choose r from n, order doesn't matter
Classical Probability	P = favorable outcomes / total outcomes	equally likely outcomes
Inclusion-Exclusion	\|A∪B\| = \|A\| + \|B\| - \|A∩B\|	union of two sets
3-Set Inclusion-Exclusion	\|A∪B∪C\| = \|A\|+\|B\|+\|C\|-\|A∩B\|-\|A∩C\|-\|B∩C\|+\|A∩B∩C\|	three sets
Independent Events	P(A∩B) = P(A) × P(B)	when A and B are independent
Addition Principle	total ways = m + n	count by categories, mutually exclusive
Multiplication Principle	total ways = m × n	count by steps, sequential

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