ACT FORMULA SHEET
Arithmetic and Algebra
Properties of Exponents and Radicals
n
__
​​  aam  ​​ = an-m
(an)m = anm
(ab)n = anbn
​​(__​  ab ​)n​​ = __
​ ​abn ​​
a-n = ​ __
​a1n  ​​
__
​ ​a1-n  ​​  = an
​​(__​  ab ​)-n
​​ = ​(​ __​  ab ​)n​​
__
1
__
​​ √   a ​​ = a​​  n  ​​
__
__ n __
​​     ab ​​ = n​​ √   a ​​ ∙ √​​    b ​​ 
√
​√ a ​ 
n _
a ___
 ​ 
√​​    b ​ ​​ = ​​  __   ​​ 
an ∙ am = an+m
n
n
n
Generic Formulas
 
_____
_
n
 
n
__
n
__
__
m
​​ √  am ​​ = a​​  n ​​  
​   b ​ 
√
2
 
Quadratic Formula: For ax2 + bx + c = 0, x = ________
​ ​-b ± ​√2ab   -4ac ​ 
​​ 
Arithmetic Mean: _____________
  
  
​ ​ Sum of Terms  ​​
Number of Terms
Event Probability: _____________
  
  
​ ​Desired Outcomes ​​
Possible Outcomes
Distance: Distance = Rate ∙ Time
Percent Growth/Decay: Original(1 ± r1)(1 ± r2)...
Percent Change: _______
​​  Old   
​​ ∙ 100%
New - Old
In Percent Growth or Decay, r1, r2, ... are the
percents an amount is being changed by each
year, month, etc.
Arithmetic Sequence/Series
Geometric Sequence/Series
Common Difference: d = an+1 - an
Common Ratio: r = ___
​ ​ an+1 ​​  
Find the nth term: an = a1 + (n - 1)d
Find the nth term: an = a1r n -1
Sum the first n terms: Sn = __
​ ​n2 ​​ (a1 + an)
Sum the first n terms: Sn = a1 ​(
​  ___  ​ ​​
​ 1 - r )
Counting and Ordering
Properties of Logarithms
Combination (Order Doesn’t Matter): nCr = ______
​ ​r!(nn!- r)!​​  
loga ax = x
a
n
1 - r n
n!
Permutation (Order Does Matter): nPr = ______
​ ​(n - r)!
   ​​ 
xloga y = loga y x
Remember, n is the number of choices you
have, and r is how many you are going to
choose.
logax + logay = loga(xy)
logax - logay = loga​​(​ _xy ​)​​
1
Geometry
The Circle
Area: A = πr2
Circumference: C = 2πr
Arc length: L(A,B) = ​ ​____
   ​​ ∙ 2πr
360°
θ
θ
Sector Area: AOB = ​ ​____
   ​​ ∙ πr 2
360°
Equation for circle with center (h, k) and radius r: (x - h)2 + (y - k)2 = r 2
Areas
Volumes
Parallelogram: A = bh
Cube: V = s3
Trapezoid: A = _​ ​12 ​​ (b1 + b2)h
Rectangular Prism: V = lwh
Triangle: A = _​ ​12 ​​ bh
Cylinder: V = πr 2 h
Cube: A = 6s 2
Sphere: V = _​ ​43 ​​πr 3
Angles
Lines
Sum of Interior Angles: = 180(n - 2)˚
Slope of a Line: m = _____
​ ​x2 - x1  ​​
180(n - 2)˚
y -y
2
1
x +x y +y
Each Interior Angle: = _______
​ ​ n  ​​  
 
Midpoint: M = ( _____
​ ​ 1 2  2 
​​,  _____
​ ​ 1 2  2 ​​ )
Sum of Exterior Angles: = 360˚
_____________
 
Distance: d = √
​  (x
  
  2 - x1)2 + (y2 - y1)2 ​​
360˚
Each Exterior Angles: = ___
​ ​ n ​​   
Pythagorean’s Theorem in 3D
d2 = a2 + b2 + c2
2
Trigonometry
Pythagorean Theorem
Pythagorean Theorem d2 = a2 + b2 + c2
Trigonometric Ratios
hypotenuse
opposite leg
csc A = _________
​ ​opposite leg 
 ​​  
adjacent leg
sec A = _________
​ ​adjacent leg 
 ​​  
opposite leg
cot A = _________
​ ​opposite leg 
 ​​
sin A = _________
​ ​hypotenuse ​​  
cos A = _________
​ ​hypotenuse ​​  
tan A = _________
​ ​adjacent leg ​​  
hypotenuse
A
b
c
adjacent leg
C
a
B
sin θ (+)
cos θ (−)
tan θ (−)
sin θ (+)
cos θ (+)
tan θ (+)
sin θ (−)
cos θ (−)
tan θ (+)
sin θ (−)
cos θ (+)
tan θ (−)
3