Formula Sheet

Browse our comprehensive collection of mathematical formulas, organized by topic. Includes quick search and interactive references for algebra, geometry, percentages, and more.

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Percentage

Percentage Formula
p=PartWhole×100%p = \frac{\text{Part}}{\text{Whole}} \times 100\%
Total Amount Formula
T=A+B+…T = A + B + \dots
Percentage Increase Formula
Vnew=Vorig×(1+p100)V_{\text{new}} = V_{\text{orig}} \times \left(1 + \frac{p}{100}\right)
Percentage Rate of Increase Formula
p=(VnewVorig−1)×100p = \left(\frac{V_{\text{new}}}{V_{\text{orig}}} - 1\right) \times 100
Percentage Decrease Formula
Vnew=Vorig×(1−p100)V_{\text{new}} = V_{\text{orig}} \times \left(1 - \frac{p}{100}\right)
General Percentage Change Formula
Vnew=Vorig×(1±p100)V_{\text{new}} = V_{\text{orig}} \times \left(1 \pm \frac{p}{100}\right)
Use + for increase and − for decrease.
Percentage Rate of Decrease Formula
p=(1−VnewVorig)×100p = \left(1 - \frac{V_{\text{new}}}{V_{\text{orig}}}\right) \times 100
General Percentage Rate Formula
p=∣VnewVorig−1∣×100p = \left|\frac{V_{\text{new}}}{V_{\text{orig}}} - 1\right| \times 100
Working Backward from Increase
Vorig=Vnew1+p100V_{\text{orig}} = \frac{V_{\text{new}}}{1 + \frac{p}{100}}
Original Value from Increase Amount
Vorig=Vinc(p100)V_{\text{orig}} = \frac{V_{\text{inc}}}{\left(\frac{p}{100}\right)}
Working Backward from Decrease
Vorig=Vnew1−p100V_{\text{orig}} = \frac{V_{\text{new}}}{1 - \frac{p}{100}}
General Working Backward Formula
Vorig=Vnew1±p100V_{\text{orig}} = \frac{V_{\text{new}}}{1 \pm \frac{p}{100}}
Use + for increase and − for decrease.
Original Value from Decrease Amount
Vorig=Vdec(p100)V_{\text{orig}} = \frac{V_{\text{dec}}}{\left(\frac{p}{100}\right)}
Total Multiplier
M=(1±p1100)×(1±p2100)×…M = \left(1 \pm \frac{p_1}{100}\right) \times \left(1 \pm \frac{p_2}{100}\right) \times \dots
New Amount
Vnew=Vorig×MV_{\text{new}} = V_{\text{orig}} \times M
Net Percentage
pnet=(M−1)×100p_{\text{net}} = (M - 1) \times 100

7th Grade Angle Fundamentals

Acute Angle
An acute angle is an angle that is greater than 0∘0^\circ and less than 90∘90^\circ.
Obtuse Angle
An obtuse angle is an angle that is greater than 90∘90^\circ and less than 180∘180^\circ.
Triangle
A triangle is a two-dimensional shape with three straight line segments called sides, and three vertices (the corner points where the sides meet).
Sum of Angles in a Triangle
a+b+c=180∘a + b + c = 180^\circ
Z-Rule (Alternate Angles)
When two parallel lines are cut by a transversal, the alternate interior angles (forming a Z shape) are equal.
F-Rule (Corresponding Angles)
When two parallel lines are cut by a transversal, the corresponding angles in the same relative positions (forming an F shape) are equal.
X-Rule (Vertically Opposite Angles)
When two straight lines intersect, the vertically opposite angles directly across from each other (forming an X shape) are equal.

7th Grade Area

Rectangle
A rectangle is a four-sided flat shape where every angle is a right angle (90∘90^\circ) and opposite sides are equal.
Area of a Rectangle
A=w×hA = w \times h
Square
A square is a four-sided flat shape with four equal sides and four right angles (90∘90^\circ).
Area of a Square
A=s2A = s^2
Area of Composite Shapes
Atotal=A1+A2+…A_{\text{total}} = A_1 + A_2 + \dots
The total area is the sum of the areas of the individual shapes.
Area of Cutouts & Holes
Aremaining=Aouter−AinnerA_{\text{remaining}} = A_{\text{outer}} - A_{\text{inner}}
The remaining area is the outer area minus the inner cutout area.
Right-Angled Triangle
A right-angled triangle is a three-sided flat shape that has one interior angle of exactly 90∘90^\circ.
Area of a Right-angled Triangle
A=ab2A = \frac{ab}{2}
Acute Triangle
An acute triangle is a triangle where all three interior angles are acute (each less than 90∘90^\circ).
Obtuse Triangle
An obtuse triangle is a triangle that has one interior angle that is obtuse (greater than 90∘90^\circ).
Height of a Triangle
The height of a triangle relative to a base is the perpendicular distance from that base (or its extension) to the opposite vertex.
Area of a Triangle
A=b×h2A = \frac{b \times h}{2}
Circle
A circle is a round flat shape where all points on the boundary are at the same distance from the center.
Area of a Circle
A=πr2A = \pi r^2
Circle Sector
A circle sector is the portion of a disk enclosed by two radii and an arc.
Area of a Circle Sector (Radians)
A=12r2θA = \frac{1}{2} r^2 \theta
Calculated using central angle θ\theta in radians.
Area of a Circle Sector (Degrees)
A=θ360∘×πr2A = \frac{\theta}{360^\circ} \times \pi r^2
Calculated using central angle θ\theta in degrees.
Angle of a Circle Sector (Radians)
θ=2Ar2\theta = \frac{2A}{r^2}
Calculated using area A and radius r in radians.
Angle of a Circle Sector (Degrees)
θ=360∘×Aπr2\theta = \frac{360^\circ \times A}{\pi r^2}
Calculated using area A and radius r in degrees.
US Customary Length Conversions
1 mi=1,760 yd1 yd=3 ft1 ft=12In1\text{ mi} = 1{,}760\text{ yd} \\ 1\text{ yd} = 3\text{ ft} \\ 1\text{ ft} = 12\text{In}
Metric Length Conversions
1 m=10 dm1 dm=10 cm1 cm=10 mm1\text{ m} = 10\text{ dm} \\ 1\text{ dm} = 10\text{ cm} \\ 1\text{ cm} = 10\text{ mm}
US & Metric Length Conversions
1In=2.54 cm1 ft=30.48 cm1 mi≈1.609 km\begin{aligned} 1\text{In} &= 2.54\text{ cm} \\ 1\text{ ft} &= 30.48\text{ cm} \\ 1\text{ mi} &\approx 1.609\text{ km} \end{aligned}
Converting between US Customary and Metric length units
Area Unit Conversion Rule
if 1 unit=k sub-unitsthen 1 unit2=k2 sub-units2\text{if}\ 1\ \text{unit} = k\ \text{sub-units} \\ \text{then}\ 1\ \text{unit}^2 = k^2\ \text{sub-units}^2
Metric Area Conversions
1 m2=100 dm21 dm2=100 cm21 cm2=100 mm21\text{ m}^2 = 100\text{ dm}^2 \\ 1\text{ dm}^2 = 100\text{ cm}^2 \\ 1\text{ cm}^2 = 100\text{ mm}^2
US Customary Area Conversions
1 mi2=3,097,600 yd21 yd2=9 ft21 ft2=144In21\text{ mi}^2 = 3{,}097{,}600\text{ yd}^2 \\ 1\text{ yd}^2 = 9\text{ ft}^2 \\ 1\text{ ft}^2 = 144\text{In}^2

Perimeter

Total Perimeter (with Holes)
Ptotal=Pouter+PinnerP_{\text{total}} = P_{\text{outer}} + P_{\text{inner}}
Perimeter of a Rectangle
P=2(w+h)P = 2(w + h)
Perimeter of a Square
P=4sP = 4s
Perimeter of a Right-angled Triangle
P=a+b+a2+b2P = a + b + \sqrt{a^2 + b^2}
Perimeter of a Polygon
P=s1+s2+⋯+snP = s_1 + s_2 + \dots + s_n
Perimeter of a Triangle
P=a+b+cP = a + b + c
Circumference of a Circle
P=2πr=πdP = 2\pi r = \pi d
Where rr is the radius, dd is the diameter (d=2rd = 2r), and π≈3.14159\pi \approx 3.14159.
Perimeter of a Circle Sector (Radians)
P=2r+θrP = 2r + \theta r
Calculated using central angle θ\theta in radians.
Perimeter of a Circle Sector (Degrees)
P=2r+θ360∘×2πrP = 2r + \frac{\theta}{360^\circ} \times 2\pi r
Calculated using central angle θ\theta in degrees.
Rectangle Diagonals Law
1. For a rectangle, the diagonals are equal.
2. Conversely, if opposite sides and diagonals are equal, it's a rectangle.

Statistics & Probability

Theoretical Probability Formula
P(A)=Number of Favorable OutcomesTotal Number of Possible OutcomesP(A) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}
Complementary Events Formula
P(A‾)=1−P(A)P(\overline{A}) = 1 - P(A)
A‾\overline{A} = not AA
Arithmetic Mean Formula
Mean=Sum of all valuesNumber of values\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}
Total Sum Formula
Total Sum=Total Count×Mean\text{Total Sum} = \text{Total Count} \times \text{Mean}
Missing Value Formula
x=Total Sum−Known Sum=Total Count×Target Mean−Known Sum\begin{aligned} x &= \text{Total Sum} - \text{Known Sum} \\ &= \text{Total Count} \times \text{Target Mean} - \text{Known Sum} \end{aligned}
Mean from Frequency Table Formula
Mean=Sum of all (Value×Count)Total Count=Grand Total SumTotal Count\text{Mean} = \frac{\text{Sum of all (Value} \times \text{Count)}}{\text{Total Count}} = \frac{\text{Grand Total Sum}}{\text{Total Count}}
Count = Frequency (how many times each value appears).
Total Count = Total Number of Items.
Definition of Median
The median is the middle value of an ordered dataset, or the average of the two middle values when the dataset has an even number of values.
Median Formula for Odd Datasets
Median (Odd n)=xn+12\text{Median (Odd n)} = x_{\frac{n+1}{2}}
nn is the number of values in the dataset.
Median Formula for Even Datasets
Median (Even n)=xn/2+x(n/2)+12\text{Median (Even n)} = \frac{x_{n/2} + x_{(n/2)+1}}{2}
nn is the number of values in the dataset.
Definition of Mode
The mode is the value (or values) that appear most frequently in a dataset. A dataset may have one mode, multiple modes, or no mode.
Range Formula (Data Spread)
Range=Maximum−Minimum=xmax⁡−xmin⁡\text{Range} = \text{Maximum} - \text{Minimum} = x_{\max} - x_{\min}

8th Grade Algebra

Expanding Binomials (Distributive Property)
(a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd
Square of a Sum
(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
Square of a Difference
(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2
Difference of Squares
(a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2
Factoring Common Factor (GCF)
ax+ay=a(x+y)ax + ay = a(x + y)
Factoring Difference of Squares
a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)
Factoring a Perfect-Square Trinomial (Sum)
a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a + b)^2
Factoring a Perfect-Square Trinomial (Difference)
a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2
Quadratic Trinomial Factoring
x2+bx+c=(x+p)(x+q)x^2 + bx + c = (x + p)(x + q)
p+q=b,p⋅q=cp + q = b, \quad p \cdot q = c

Exponents

Product Rule of Exponents
am⋅an=am+na^m \cdot a^n = a^{m+n}
Quotient Rule of Exponents
aman=am−n\frac{a^m}{a^n} = a^{m-n}
Power of a Power Rule
(am)n=am⋅n(a^m)^n = a^{m \cdot n}
Power of a Product Rule
(a⋅b)n=an⋅bn(a \cdot b)^n = a^n \cdot b^n
Power of a Quotient Rule
(ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
Zero Exponent Rule
a0=1(a≠0)a^0 = 1 \quad (a \neq 0)
Negative Exponent Rule
a−n=1an(a≠0)a^{-n} = \frac{1}{a^n} \quad (a \neq 0)
Exponent −1 Rule
a−1=1a(a≠0)a^{-1} = \frac{1}{a} \quad (a \neq 0)
Negative Exponent Rule for Fractions
(ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n
Definition of Square Root
x2=a  ⟺  x=±a(a≥0)x^2 = a \iff x = \pm\sqrt{a} \quad (a \ge 0)
Definition of Cube Root
x3=a  ⟺  x=a3x^3 = a \iff x = \sqrt[3]{a}
Definition of Scientific Notation
a×10k(1≤a<10,  k∈Z)a \times 10^k \quad (1 \le a < 10,\; k \in \mathbb{Z})
Multiplication in Scientific Notation
(a×10m)×(b×10n)=(a⋅b)×10m+n(a \times 10^m) \times (b \times 10^n) = (a \cdot b) \times 10^{m+n}
Division in Scientific Notation
a×10mb×10n=(ab)×10m−n\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}
Metric Prefixes
tera (T)=1012giga (G)=109mega (M)=106kilo (k)=103centi (c)=10−2milli (m)=10−3micro (μ)=10−6nano (n)=10−9pico (p)=10−12\begin{aligned} \text{tera}\text{ (T)} &= 10^{12} \\ \text{giga}\text{ (G)} &= 10^9 \\ \text{mega}\text{ (M)} &= 10^6 \\ \text{kilo}\text{ (k)} &= 10^3 \\ \text{centi}\text{ (c)} &= 10^{-2} \\ \text{milli}\text{ (m)} &= 10^{-3} \\ \text{micro}\text{ (}\mu\text{)} &= 10^{-6} \\ \text{nano}\text{ (n)} &= 10^{-9} \\ \text{pico}\text{ (p)} &= 10^{-12} \end{aligned}
Scale factors relative to the base unit

Functions

Coordinates of a Point
(x,y)(x, y)
x: horizontal position
y: vertical position
Midpoint coordinates of a segment
M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
A: (x₁, y₁), B: (x₂, y₂)
Distance between two points
d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
d: Distance, A(x₁, y₁), B(x₂, y₂)
Continuity of a Function
A function f(x)f(x) is continuous at a point x=ax = a if:
1. f(a)f(a) is defined.
2. The graph approaches the same yy-value from both the left and the right.
3. The function value at aa matches this yy-value (no jumps or holes).
Increasing & Decreasing Functions
A function ff is:
  • Increasing if f(x1)<f(x2)f(x_1) < f(x_2) for all x1<x2x_1 < x_2 in its entire domain (always rises as xx increases).
  • Decreasing if f(x1)>f(x2)f(x_1) > f(x_2) for all x1<x2x_1 < x_2 in its entire domain (always falls as xx increases).
Intervals of Increase & Decrease
A function increases or decreases on a specific interval II if the relationship holds for all inputs in II:
  • Increasing on II: f(x1)<f(x2)f(x_1) < f(x_2) for all x1<x2x_1 < x_2 in II.
  • Decreasing on II: f(x1)>f(x2)f(x_1) > f(x_2) for all x1<x2x_1 < x_2 in II.

8th Grade Triangle Congruence

Definition of Congruent Triangles
ΔABC≅ΔDEF  ⟺  {AB=DE,BC=EF,AC=DF∠A=∠D,∠B=∠E,∠C=∠F\Delta ABC \cong \Delta DEF \iff \begin{cases} AB = DE, & BC = EF, & AC = DF \\ \angle A = \angle D, & \angle B = \angle E, & \angle C = \angle F \end{cases}
Side-Angle-Side Congruence Theorem (SAS)
AB=DE,  ∠B=∠E,  BC=EF  ⟹  ΔABC≅ΔDEFAB = DE,\; \angle B = \angle E,\; BC = EF \implies \Delta ABC \cong \Delta DEF
Angle-Side-Angle Congruence Theorem (ASA)
∠A=∠D,  AB=DE,  ∠B=∠E  ⟹  ΔABC≅ΔDEF\angle A = \angle D,\; AB = DE,\; \angle B = \angle E \implies \Delta ABC \cong \Delta DEF
Side-Side-Side Congruence Theorem (SSS)
AB=DE,  BC=EF,  AC=DF  ⟹  ΔABC≅ΔDEFAB = DE,\; BC = EF,\; AC = DF \implies \Delta ABC \cong \Delta DEF
Special Side-Side-Angle (SSA) Case (Angle Opposite Longer Side) / Hypotenuse-Leg (HL)
AB=DE,  BC=EF,  ∠C=∠F=90∘  ⟹  ΔABC≅ΔDEFAB = DE,\; BC = EF,\; \angle C = \angle F = 90^\circ \implies \Delta ABC \cong \Delta DEF
Definition & Property of Triangle Median
In  ΔABC (D  on  BC):AD  is a median  ⟺  BD=DC=12BC\begin{gathered} \text{In}\; \Delta ABC \ (D\; \text{on}\; BC): \\ AD\; \text{is a median} \iff BD = DC = \frac{1}{2} BC \end{gathered}
Definition & Property of Triangle Altitude
In  ΔABC (H  on (or extension of)  BC):AH  is an altitude  ⟺  AH⊥BC  (∠AHB=90∘)\begin{gathered} \text{In}\; \Delta ABC \ (H\; \text{on (or extension of)}\; BC): \\ AH\; \text{is an altitude} \iff AH \perp BC \; (\angle AHB = 90^\circ) \end{gathered}
Definition & Property of Triangle Angle Bisector
In  ΔABC (D  on  BC):AD  bisects  ∠BAC  ⟺  ∠BAD=∠CAD=12∠BAC\begin{gathered} \text{In}\; \Delta ABC \ (D\; \text{on}\; BC): \\ AD\; \text{bisects}\; \angle BAC \iff \angle BAD = \angle CAD = \frac{1}{2} \angle BAC \end{gathered}
Triangle Midsegment Theorem
In  ΔABC (M  on  AB,  N  on  AC):MN  is a midsegment  ⟺  MN∥BC,  MN=12BC\begin{gathered} \text{In}\; \Delta ABC \ (M\; \text{on}\; AB,\; N\; \text{on}\; AC): \\ MN\; \text{is a midsegment} \iff MN \parallel BC,\; MN = \frac{1}{2} BC \end{gathered}
Isosceles Triangle Base Angles Theorem & Converse
AB=AC  ⟺  ∠B=∠CAB = AC \iff \angle B = \angle C
Isosceles Triangle Special Line (Bisector = Median = Altitude)
In isosceles  ΔABC (AB=AC):Angle Bisector  ⟺  Median  ⟺  Altitude\begin{gathered} \text{In isosceles}\; \Delta ABC \ (AB=AC): \\ \text{Angle Bisector} \iff \text{Median} \iff \text{Altitude} \end{gathered}
Isometry (Rigid Motion Congruence)
d(A′,B′)=d(A,B)  ⟹  ΔABC≅ΔA′B′C′d(A', B') = d(A, B) \implies \Delta ABC \cong \Delta A'B'C'
Translation (Shift Vector)
(x,y)→(x+a,y+b)(x, y) \to (x + a, y + b)
Reflection over the x-Axis
(x,y)→reflect x-axis(x,−y)(x, y) \xrightarrow{\text{reflect x-axis}} (x, -y)
Reflection over the y-Axis
(x,y)→reflect y-axis(−x,y)(x, y) \xrightarrow{\text{reflect y-axis}} (-x, y)
Reflection over the Line y = x
(x,y)→reflect y = x(y,x)(x, y) \xrightarrow{\text{reflect y = x}} (y, x)
Rotation 90° Counterclockwise about Origin
(x,y)→R(O,90∘CCW)(−y,x)(x, y) \xrightarrow{R_{(O, 90^\circ \text{CCW})}} (-y, x)
Rotation 180° about Origin
(x,y)→R(O,180∘)(−x,−y)(x, y) \xrightarrow{R_{(O, 180^\circ)}} (-x, -y)
Rotation 270° Counterclockwise (90° Clockwise) about Origin
(x,y)→R(O,270∘CCW)(y,−x)(x, y) \xrightarrow{R_{(O, 270^\circ \text{CCW})}} (y, -x)
Definition of Similar Triangles (~)
ΔABC∼ΔDEF  ⟺  {∠A=∠D,  ∠B=∠E,  ∠C=∠FABDE=BCEF=ACDF=k\Delta ABC \sim \Delta DEF \iff \begin{cases} \angle A = \angle D, \; \angle B = \angle E, \; \angle C = \angle F \\[4pt] \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k \end{cases}
Dilation on the Coordinate Plane (Scale Factor k)
(x,y)→dilation k(k⋅x,k⋅y)(x, y) \xrightarrow{\text{dilation } k} (k \cdot x, k \cdot y)
Angle-Angle Similarity Criterion (AA)
∠A=∠D,  ∠B=∠E  ⟹  ΔABC∼ΔDEF\angle A = \angle D, \; \angle B = \angle E \implies \Delta ABC \sim \Delta DEF
Side-Angle-Side Similarity Criterion (SAS ~)
ABDE=ACDF=k,  ∠A=∠D  ⟹  ΔABC∼ΔDEF\frac{AB}{DE} = \frac{AC}{DF} = k, \; \angle A = \angle D \implies \Delta ABC \sim \Delta DEF
Side-Side-Side Similarity Criterion (SSS ~)
ABDE=BCEF=ACDF=k  ⟹  ΔABC∼ΔDEF\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k \implies \Delta ABC \sim \Delta DEF
Triangle Proportionality Theorem (Parallel Lines in Triangles)
DE∥BC  ⟹  ADAB=AEAC=DEBCDE \parallel BC \implies \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}
Ratios of Perimeters and Areas of Similar Figures
P1P2=k,S1S2=k2\frac{P_1}{P_2} = k, \quad \frac{S_1}{S_2} = k^2

8th Grade Statistics & Data Analysis

Slope of Trend Line / Line of Best Fit
m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
mm is the slope / rate of change, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are points on the line
Equation of Line of Best Fit
y=mx+by = mx + b
mm is the rate of change, bb is the initial value / y-intercept
Two-Way Table Relative Frequency
Relative Frequency=Joint FrequencyTotal\text{Relative Frequency} = \frac{\text{Joint Frequency}}{\text{Total}}
Five-Number Summary
Summary={Minimum, Q1, Median, Q3, Maximum}\text{Summary} = \{\text{Minimum},\, Q_1,\, \text{Median},\, Q_3,\, \text{Maximum}\}
Min\text{Min} is minimum, Q1Q_1 is first quartile, Median\text{Median} is second quartile, Q3Q_3 is third quartile, Max\text{Max} is maximum
Interquartile Range (IQR)
IQR=Q3−Q1\text{IQR} = Q_3 - Q_1
Q1Q_1 is first quartile, Q3Q_3 is third quartile
Outlier Identification Rule (1.5 × IQR Rule)
Outlier<Q1−1.5×IQRorOutlier>Q3+1.5×IQR\begin{gathered} \text{Outlier} < Q_1 - 1.5 \times \text{IQR} \\[4pt] \text{or} \\[4pt] \text{Outlier} > Q_3 + 1.5 \times \text{IQR} \end{gathered}

8th Grade Area & Pythagoras

Pythagorean Theorem
a2+b2=c2a^2 + b^2 = c^2
For a right-angled triangle: the square of the hypotenuse (c) is equal to the sum of the squares of the legs (a and b).
Altitude of an Isosceles Triangle
h=b2−(a2)2h = \sqrt{b^2 - \left(\frac{a}{2}\right)^2}
Equilateral Triangle Law
1. If a triangle is equilateral, all its angles are 60∘60^\circ.
2. Conversely, if all angles of a triangle are 60∘60^\circ, the triangle is equilateral.
Height of an Equilateral Triangle
h=a32h = a \frac{\sqrt{3}}{2}
Area of an Equilateral Triangle
A=a234A = a^2 \frac{\sqrt{3}}{4}
30-60-90 Triangle Law
In any 30-60-90 triangle, the leg opposite the 30∘30^\circ angle (the shortest leg) is always half the length of the hypotenuse: a=c2a = \frac{c}{2} (or c=2ac = 2a).
Area of a 30-60-90 Triangle
A=a232A = a^2 \frac{\sqrt{3}}{2}
where a is the shortest leg.
Ratio of Inscribed Square to Circle
AsquareAcircle=2π\frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{2}{\pi}
Ratio of Inscribed Circle to Square
AcircleAsquare=π4\frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi}{4}

Volume & Surface Area

Box (Rectangular Prism)
A box (rectangular prism) is a three-dimensional solid with six rectangular faces, where opposite faces are equal and parallel.
Volume of a Box
V=l×w×hV = l \times w \times h
Cube
A cube is a three-dimensional solid bounded by six equal square faces.
Volume of a Cube
V=s3V = s^3
Volume of Composite Shapes
Vtotal=V1+V2+…V_{\text{total}} = V_1 + V_2 + \dots
When boxes are stacked or placed side-by-side, the total volume is the sum of the volumes of the individual boxes.
Volume of Cutouts & Holes
Vremaining=Vouter−VinnerV_{\text{remaining}} = V_{\text{outer}} - V_{\text{inner}}
If a smaller box or cube is cut out from a larger box, the remaining volume is the outer volume minus the inner cutout volume.
Surface Area of a Box
A=2(l×w+l×h+w×h)A = 2(l \times w + l \times h + w \times h)
Surface Area of a Cube
A=6s2A = 6s^2
Surface Area of Composite Shapes
Atotal=A1+A2−2×AoverlapA_{\text{total}} = A_1 + A_2 - 2 \times A_{\text{overlap}}
When boxes share a face, subtract twice the overlapping area from the sum of their individual surface areas.
Surface Area with a Cutout
Aremaining=Aouter−2(winner×hinner)+2(louter×hinner)\begin{aligned} A_{\text{remaining}} &= A_{\text{outer}} - 2(w_{\text{inner}} \times h_{\text{inner}}) \\ &\quad + 2(l_{\text{outer}} \times h_{\text{inner}}) \end{aligned}
The surface area of a box with a channel cutout is the outer surface area minus twice the removed front/back notch area plus twice the new inner wall area.
Cylinder
A cylinder is a three-dimensional solid with two parallel, congruent circular bases connected by a curved surface.
Volume of a Cylinder
V=πr2hV = \pi r^2 h
Cylinder Sector (Wedge)
A cylinder sector prism (cylinder wedge) is a fraction of a full cylinder.
Volume of a Cylinder Sector
V=θ360∘πr2hV = \frac{\theta}{360^\circ} \pi r^2 h
Calculated using radius rr, angle θ\theta in degrees, and height hh.
Surface Area of a Cylinder
A=2πr2+2πrhA = 2\pi r^2 + 2\pi r h
Surface Area of a Cylinder Sector
A=2×(θ360∘πr2)+2rh+θ180∘πrhA = 2 \times \left(\frac{\theta}{360^\circ} \pi r^2\right) + 2 r h + \frac{\theta}{180^\circ} \pi r h
Calculated using base radius rr, central angle θ\theta in degrees, and height hh.
Prism
A prism is a three-dimensional solid with a constant cross-section (base).
Volume of a General Prism
V=Base Area×heightV = \text{Base Area} \times \text{height}
A prism is a three-dimensional solid with a constant cross-section (base). The volume of any prism is calculated by multiplying the area of its base by its height.
Surface Area of a General Prism
A=2×Base Area+Perimeter×heightA = 2 \times \text{Base Area} + \text{Perimeter} \times \text{height}
Sphere
A sphere is a perfectly round three-dimensional solid where every point on the surface is equidistant from the center.
Volume of a Sphere
V=43πr3V = \frac{4}{3} \pi r^3
Spherical Shell
A spherical shell is a three-dimensional solid bounded by two concentric spheres of different radii.
Volume of a Spherical Shell
V=43π(R3−r3)V = \frac{4}{3} \pi \left(R^3 - r^3\right)
Calculated using outer radius RR and inner radius rr.
Surface Area of a Sphere
A=4πr2A = 4 \pi r^2
Surface Area of a Spherical Shell
A=4π(R2+r2)A = 4 \pi \left(R^2 + r^2\right)
Calculated as the sum of outer and inner surface areas using outer radius RR and inner radius rr.
Pyramid
A pyramid is a three-dimensional solid with a polygonal base and triangular lateral faces that meet at a single point called the apex.
Right Pyramid
A pyramid where the apex is vertically aligned directly above the geometric center of the base.
Oblique Pyramid
A pyramid where the apex is shifted and not vertically aligned above the base center. The perpendicular height drops to the plane containing the base outside its center.
Volume of a Pyramid (Square)
V=13s2hV = \frac{1}{3} s^2 h
A pyramid is a three-dimensional solid with a polygonal base and triangular lateral faces that meet at a single point called the apex.
Cone
A cone is a three-dimensional solid with a circular base that tapers smoothly to a single point called the apex.
Volume of a Cone
V=13πr2hV = \frac{1}{3} \pi r^2 h
Volume of a Pyramid or Cone (General)
V=13BhV = \frac{1}{3} B h
Pyramids and cones have the same general volume formula: one-third of the base area times the height.
Surface Area of a Pyramid (Square)
A=s2+2slA = s^2 + 2 s l
Where ss is the base side length and ll is the slant height.
Slant Height of a Pyramid
l=h2+(s2)2l = \sqrt{h^2 + \left(\frac{s}{2}\right)^2}
Where hh is the perpendicular height and ss is the base side length.
Surface Area of a Cone
A=πr2+πrlA = \pi r^2 + \pi r l
Where rr is the base radius and ll is the slant height.
Slant Height of a Cone
l=h2+r2l = \sqrt{h^2 + r^2}
Where rr is the base radius and hh is the perpendicular height.
Volume Unit Conversion Rule
if 1 unit=k sub-units  ⟹  1 unit3=k3 sub-units3\text{if}\ 1\ \text{unit} = k\ \text{sub-units} \implies 1\ \text{unit}^3 = k^3\ \text{sub-units}^3
Metric Volume Conversions
1 m3=1,000 dm31 dm3=1,000 cm31 cm3=1,000 mm31\text{ m}^3 = 1{,}000\text{ dm}^3 \\ 1\text{ dm}^3 = 1{,}000\text{ cm}^3 \\ 1\text{ cm}^3 = 1{,}000\text{ mm}^3
US Customary Volume Conversions
1 yd3=27 ft31 ft3=1,728In31\text{ yd}^3 = 27\text{ ft}^3 \\ 1\text{ ft}^3 = 1{,}728\text{In}^3
Scaling Dimensions Rules
Linear (1D) scales by kk
Area (2D) scales by k2k^2
Volume (3D) scales by k3k^3