ACT Math Formula Sheet — Complete Reference for Test Day A complete reference of every math formula you need for the ACT. Unlike the SAT, the ACT does NOT provide a formula sheet on test day — everything here you should know cold. Covers the six ACT math content areas plus the test's most common trap topics.
Pre-Algebra & Elementary Algebra Order of Operations
PEMDAS: Parens, Exponents, ×÷, +− \text{PEMDAS: Parens, Exponents, ×÷, +−} PEMDAS: Parens, Exponents, ×÷, +− Percent Change
new − old old × 100 % \frac{\text{new} - \text{old}}{\text{old}} \times 100\% old new − old × 100% Percent of
x % of y = x 100 ⋅ y x\% \text{ of } y = \tfrac{x}{100} \cdot y x % of y = 100 x ⋅ y Average (Mean)
x ˉ = ∑ x i n \bar{x} = \frac{\sum x_i}{n} x ˉ = n ∑ x i Ratio
a : b = a b a : b = \frac{a}{b} a : b = b a Probability
P = favorable total P = \frac{\text{favorable}}{\text{total}} P = total favorable Exponents & Roots Product of Powers
a m a n = a m + n a^m a^n = a^{m+n} a m a n = a m + n Quotient of Powers
a m a n = a m − n \tfrac{a^m}{a^n} = a^{m-n} a n a m = a m − n Power of a Power
( a m ) n = a m n (a^m)^n = a^{mn} ( a m ) n = a mn Negative Exponent
a − n = 1 a n a^{-n} = \tfrac{1}{a^n} a − n = a n 1 Fractional Exponent
a m / n = a m n a^{m/n} = \sqrt[n]{a^m} a m / n = n a m Zero Exponent
a 0 = 1 ( a ≠ 0 ) a^0 = 1 \quad(a \ne 0) a 0 = 1 ( a = 0 ) Intermediate Algebra Quadratic Formula
x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c Discriminant
Δ = b 2 − 4 a c \Delta = b^2 - 4ac Δ = b 2 − 4 a c Sum of Roots
r 1 + r 2 = − b a r_1 + r_2 = -\frac{b}{a} r 1 + r 2 = − a b Product of Roots
r 1 r 2 = c a r_1 r_2 = \frac{c}{a} r 1 r 2 = a c Difference of Squares
a 2 − b 2 = ( a + b ) ( a − b ) a^2 - b^2 = (a + b)(a - b) a 2 − b 2 = ( a + b ) ( a − b ) Perfect Square
( a + b ) 2 = a 2 + 2 a b + b 2 (a + b)^2 = a^2 + 2ab + b^2 ( a + b ) 2 = a 2 + 2 ab + b 2 Function Notation
f ( g ( x ) ) = ( f ∘ g ) ( x ) f(g(x)) = (f \circ g)(x) f ( g ( x )) = ( f ∘ g ) ( x ) Coordinate Geometry Slope
m = y 2 − y 1 x 2 − x 1 m = \frac{y_2 - y_1}{x_2 - x_1} m = x 2 − x 1 y 2 − y 1 Point-Slope Form
y − y 1 = m ( x − x 1 ) y - y_1 = m(x - x_1) y − y 1 = m ( x − x 1 ) Distance Formula
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Midpoint
M = ( x 1 + x 2 2 , y 1 + y 2 2 ) M = \left(\tfrac{x_1 + x_2}{2}, \tfrac{y_1 + y_2}{2}\right) M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) Perpendicular Lines
m 1 m 2 = − 1 m_1 m_2 = -1 m 1 m 2 = − 1 Circle Equation
( x − h ) 2 + ( y − k ) 2 = r 2 (x - h)^2 + (y - k)^2 = r^2 ( x − h ) 2 + ( y − k ) 2 = r 2 Plane Geometry Pythagorean Theorem
a 2 + b 2 = c 2 a^2 + b^2 = c^2 a 2 + b 2 = c 2 Triangle Area
A = 1 2 b h A = \tfrac{1}{2} b h A = 2 1 bh Triangle Angle Sum
A + B + C = 180 ° A + B + C = 180° A + B + C = 180° Trapezoid Area
A = 1 2 ( b 1 + b 2 ) h A = \tfrac{1}{2}(b_1 + b_2)h A = 2 1 ( b 1 + b 2 ) h Polygon Angle Sum
S = ( n − 2 ) ⋅ 180 ° S = (n - 2) \cdot 180° S = ( n − 2 ) ⋅ 180° Solid Geometry Cylinder Volume
V = π r 2 h V = \pi r^2 h V = π r 2 h Cone Volume
V = 1 3 π r 2 h V = \tfrac{1}{3} \pi r^2 h V = 3 1 π r 2 h Sphere Volume
V = 4 3 π r 3 V = \tfrac{4}{3} \pi r^3 V = 3 4 π r 3 Sphere Surface Area
S A = 4 π r 2 SA = 4 \pi r^2 S A = 4 π r 2 Special Right Triangles 45-45-90
sides 1 : 1 : 2 \text{sides } 1 : 1 : \sqrt{2} sides 1 : 1 : 2 30-60-90
sides 1 : 3 : 2 \text{sides } 1 : \sqrt{3} : 2 sides 1 : 3 : 2 Trigonometry SOHCAHTOA
sin = opp hyp , cos = adj hyp , tan = opp adj \sin = \tfrac{\text{opp}}{\text{hyp}},\ \cos = \tfrac{\text{adj}}{\text{hyp}},\ \tan = \tfrac{\text{opp}}{\text{adj}} sin = hyp opp , cos = hyp adj , tan = adj opp Pythagorean Identity
sin 2 θ + cos 2 θ = 1 \sin^2 \theta + \cos^2 \theta = 1 sin 2 θ + cos 2 θ = 1 Tangent Identity
tan θ = sin θ cos θ \tan \theta = \frac{\sin \theta}{\cos \theta} tan θ = cos θ sin θ Law of Sines
a sin A = b sin B = c sin C \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} sin A a = sin B b = sin C c Law of Cosines
c 2 = a 2 + b 2 − 2 a b cos C c^2 = a^2 + b^2 - 2ab \cos C c 2 = a 2 + b 2 − 2 ab cos C Reciprocal Identities
csc = 1 sin , sec = 1 cos , cot = 1 tan \csc = \tfrac{1}{\sin},\ \sec = \tfrac{1}{\cos},\ \cot = \tfrac{1}{\tan} csc = s i n 1 , sec = c o s 1 , cot = t a n 1 