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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Ratio

Proportional Relationship Formula

y = kx

Two quantities x and y are proportional if the ratio y/x is constant and equal to k.

Constant of Proportionality (k)

k = \frac{y}{x}

The constant ratio y/x (unit rate) in a proportional relationship.

Percentage

Percentage Formula

p = \frac{\text{Part}}{\text{Whole}} \times 100\%

Total Amount Formula

T = A + B + \dots

Percentage Increase Formula

V_{\text{new}} = V_{\text{orig}} \times \left(1 + \frac{p}{100}\right)

Percentage Rate of Increase Formula

p = \left(\frac{V_{\text{new}}}{V_{\text{orig}}} - 1\right) \times 100

Percentage Decrease Formula

V_{\text{new}} = V_{\text{orig}} \times \left(1 - \frac{p}{100}\right)

General Percentage Change Formula

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 = \left(1 - \frac{V_{\text{new}}}{V_{\text{orig}}}\right) \times 100

General Percentage Rate Formula

p = \left|\frac{V_{\text{new}}}{V_{\text{orig}}} - 1\right| \times 100

Working Backward from Increase

V_{\text{orig}} = \frac{V_{\text{new}}}{1 + \frac{p}{100}}

Original Value from Increase Amount

V_{\text{orig}} = \frac{V_{\text{inc}}}{\left(\frac{p}{100}\right)}

Working Backward from Decrease

V_{\text{orig}} = \frac{V_{\text{new}}}{1 - \frac{p}{100}}

General Working Backward Formula

V_{\text{orig}} = \frac{V_{\text{new}}}{1 \pm \frac{p}{100}}

Use + for increase and − for decrease.

Original Value from Decrease Amount

V_{\text{orig}} = \frac{V_{\text{dec}}}{\left(\frac{p}{100}\right)}

Total Multiplier

M = \left(1 \pm \frac{p_1}{100}\right) \times \left(1 \pm \frac{p_2}{100}\right) \times \dots

New Amount

V_{\text{new}} = V_{\text{orig}} \times M

Net Percentage

p_{\text{net}} = (M - 1) \times 100

7th Grade Area

Rectangle

A rectangle is a four-sided flat shape where every angle is a right angle (

90^\circ

) and opposite sides are equal.

Area of a Rectangle

A = w \times h

Square

A square is a four-sided flat shape with four equal sides and four right angles (

90^\circ

Area of a Square

A = s^2

Area of Composite Shapes

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

A_{\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^\circ

Area of a Right-angled Triangle

A = \frac{ab}{2}

Acute Triangle

An acute triangle is a triangle where all three interior angles are acute (each less than

90^\circ

Obtuse Triangle

An obtuse triangle is a triangle that has one interior angle that is obtuse (greater than

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.

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 = \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 = \frac{1}{2} r^2 \theta

Calculated using central angle

\theta

in radians.

Area of a Circle Sector (Degrees)

A = \frac{\theta}{360^\circ} \times \pi r^2

Calculated using central angle

\theta

in degrees.

Angle of a Circle Sector (Radians)

\theta = \frac{2A}{r^2}

Calculated using area A and radius r in radians.

Angle of a Circle Sector (Degrees)

\theta = \frac{360^\circ \times A}{\pi r^2}

Calculated using area A and radius r in degrees.

Area of a Triangle

A = \frac{b \times h}{2}

US Customary Length Conversions

1\text{ mi} = 1{,}760\text{ yd} \\ 1\text{ yd} = 3\text{ ft} \\ 1\text{ ft} = 12\text{ in}

Metric Length Conversions

1\text{ m} = 10\text{ dm} \\ 1\text{ dm} = 10\text{ cm} \\ 1\text{ cm} = 10\text{ mm}

Metric Prefixes

\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

US & Metric Length Conversions

\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

\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\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\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)

P_{\text{total}} = P_{\text{outer}} + P_{\text{inner}}

Perimeter of a Rectangle

P = 2(w + h)

Perimeter of a Square

P = 4s

Perimeter of a Right-angled Triangle

P = a + b + \sqrt{a^2 + b^2}

Perimeter of a Polygon

P = s_1 + s_2 + \dots + s_n

Perimeter of a Triangle

P = a + b + c

Circumference of a Circle

P = 2\pi r = \pi d

Where

r

is the radius,

d

is the diameter (

d = 2r

), and

\pi \approx 3.14159

Perimeter of a Circle Sector (Radians)

P = 2r + \theta r

Calculated using central angle

\theta

in radians.

Perimeter of a Circle Sector (Degrees)

P = 2r + \frac{\theta}{360^\circ} \times 2\pi r

Calculated using central angle

\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.

7th Grade Angle Fundamentals

Acute Angle

An acute angle is an angle that is greater than

0^\circ

and less than

90^\circ

Obtuse Angle

An obtuse angle is an angle that is greater than

90^\circ

and less than

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^\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.

Linear Equations

Linear Equation (Slope-Intercept Form)

y = ax + b

The slope-intercept form of a linear equation, where a represents the slope (also called the gradient) and b represents the y-intercept.

Slope of a Line

a = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

Calculated from two points

(x_1, y_1)

and

(x_2, y_2)

on the line.

Property of Parallel Lines

The graphs of two linear functions are parallel if and only if they have the same slope:

a_1 = a_2

(and different y-intercepts:

b_1 \neq b_2

Functions

Coordinates of a Point

(x, y)

x: horizontal position
y: vertical position

Midpoint coordinates of a segment

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 = \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)

is continuous at a point

x = a

if:
1.

f(a)

is defined.
2. The graph approaches the same

y

-value from both the left and the right.
3. The function value at

a

matches this

y

-value (no jumps or holes).

Increasing & Decreasing Functions

A function

f

is:
• Increasing if

f(x_1) < f(x_2)

for all

x_1 < x_2

in its entire domain (always rises as

x

increases).
• Decreasing if

f(x_1) > f(x_2)

for all

x_1 < x_2

in its entire domain (always falls as

x

increases).

Intervals of Increase & Decrease

A function increases or decreases on a specific interval

I

if the relationship holds for all inputs in

I

:
• Increasing on $I$ :

f(x_1) < f(x_2)

for all

x_1 < x_2

I

.
• Decreasing on $I$ :

f(x_1) > f(x_2)

for all

x_1 < x_2

I

8th Grade Area & Pythagoras

Pythagorean Theorem

a^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 = \sqrt{b^2 - \left(\frac{a}{2}\right)^2}

Equilateral Triangle Law

1. If a triangle is equilateral, all its angles are

60^\circ

.
2. Conversely, if all angles of a triangle are

60^\circ

, the triangle is equilateral.

Height of an Equilateral Triangle

h = a \frac{\sqrt{3}}{2}

Area of an Equilateral Triangle

A = a^2 \frac{\sqrt{3}}{4}

30-60-90 Triangle Law

In any 30-60-90 triangle, the leg opposite the

30^\circ

angle (the shortest leg) is always half the length of the hypotenuse:

a = \frac{c}{2}

(or

c = 2a

Area of a 30-60-90 Triangle

A = a^2 \frac{\sqrt{3}}{2}

where a is the shortest leg.

Ratio of Inscribed Square to Circle

\frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{2}{\pi}

Ratio of Inscribed Circle to Square

\frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi}{4}

8th Grade Angles & Radians

Convert Degrees to Radians

\text{rad} = \text{deg} \times \frac{\pi}{180^\circ}

Convert Radians to Degrees

\text{deg} = \text{rad} \times \frac{180^\circ}{\pi}

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 \times w \times h

Cube

A cube is a three-dimensional solid bounded by six equal square faces.

Volume of a Cube

V = s^3

Volume of Composite Shapes

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

V_{\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 \times w + l \times h + w \times h)

Surface Area of a Cube

A = 6s^2

Surface Area of Composite Shapes

A_{\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

\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 = \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 = \frac{\theta}{360^\circ} \pi r^2 h

Calculated using radius

r

, angle

\theta

in degrees, and height

h

Surface Area of a Cylinder

A = 2\pi r^2 + 2\pi r h

Surface Area of a Cylinder Sector

A = 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

r

, central angle

\theta

in degrees, and height

h

Prism

A prism is a three-dimensional solid with a constant cross-section (base).

Volume of a General Prism

V = \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 \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 = \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 = \frac{4}{3} \pi \left(R^3 - r^3\right)

Calculated using outer radius

R

and inner radius

r

Surface Area of a Sphere

A = 4 \pi r^2

Surface Area of a Spherical Shell

A = 4 \pi \left(R^2 + r^2\right)

Calculated as the sum of outer and inner surface areas using outer radius

R

and inner radius

r

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 = \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 = \frac{1}{3} \pi r^2 h

Volume of a Pyramid or Cone (General)

V = \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 = s^2 + 2 s l

Where

s

is the base side length and

l

is the slant height.

Slant Height of a Pyramid

l = \sqrt{h^2 + \left(\frac{s}{2}\right)^2}

Where

h

is the perpendicular height and

s

is the base side length.

Surface Area of a Cone

A = \pi r^2 + \pi r l

Where

r

is the base radius and

l

is the slant height.

Slant Height of a Cone

l = \sqrt{h^2 + r^2}

Where

r

is the base radius and

h

is the perpendicular height.

Volume Unit Conversion Rule

\text{if}\ 1\ \text{unit} = k\ \text{sub-units} \implies 1\ \text{unit}^3 = k^3\ \text{sub-units}^3

Metric Volume Conversions

1\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\text{ yd}^3 = 27\text{ ft}^3 \\ 1\text{ ft}^3 = 1{,}728\text{ in}^3

Scaling Dimensions Rules

Linear (1D) scales by

k

Area (2D) scales by

k^2

Volume (3D) scales by

k^3

Formula Sheet

List of all mathematical formulas on the site, organized by topic.

Ratio

Proportional Relationship Formula

y = kx

Two quantities x and y are proportional if the ratio y/x is constant and equal to k.

Constant of Proportionality (k)

k = \frac{y}{x}

The constant ratio y/x (unit rate) in a proportional relationship.

Percentage

Percentage Formula

p = \frac{\text{Part}}{\text{Whole}} \times 100\%

Total Amount Formula

T = A + B + \dots

Percentage Increase Formula

V_{\text{new}} = V_{\text{orig}} \times \left(1 + \frac{p}{100}\right)

Percentage Rate of Increase Formula

p = \left(\frac{V_{\text{new}}}{V_{\text{orig}}} - 1\right) \times 100

Percentage Decrease Formula

V_{\text{new}} = V_{\text{orig}} \times \left(1 - \frac{p}{100}\right)

General Percentage Change Formula

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 = \left(1 - \frac{V_{\text{new}}}{V_{\text{orig}}}\right) \times 100

General Percentage Rate Formula

p = \left|\frac{V_{\text{new}}}{V_{\text{orig}}} - 1\right| \times 100

Working Backward from Increase

V_{\text{orig}} = \frac{V_{\text{new}}}{1 + \frac{p}{100}}

Original Value from Increase Amount

V_{\text{orig}} = \frac{V_{\text{inc}}}{\left(\frac{p}{100}\right)}

Working Backward from Decrease

V_{\text{orig}} = \frac{V_{\text{new}}}{1 - \frac{p}{100}}

General Working Backward Formula

V_{\text{orig}} = \frac{V_{\text{new}}}{1 \pm \frac{p}{100}}

Use + for increase and − for decrease.

Original Value from Decrease Amount

V_{\text{orig}} = \frac{V_{\text{dec}}}{\left(\frac{p}{100}\right)}

Total Multiplier

M = \left(1 \pm \frac{p_1}{100}\right) \times \left(1 \pm \frac{p_2}{100}\right) \times \dots

New Amount

V_{\text{new}} = V_{\text{orig}} \times M

Net Percentage

p_{\text{net}} = (M - 1) \times 100

7th Grade Area

Rectangle

formula_definition_rectangle_text

Area of a Rectangle

A = w \times h

Square

formula_definition_square_text

Area of a Square

A = s^2

Area of Composite Shapes

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

A_{\text{remaining}} = A_{\text{outer}} - A_{\text{inner}}

The remaining area is the outer area minus the inner cutout area.

Right-Angled Triangle

formula_definition_right_triangle_text

Area of a Right-angled Triangle

A = \frac{ab}{2}

Acute Triangle

formula_definition_acute_triangle_text

Obtuse Triangle

formula_definition_obtuse_triangle_text

Height of a Triangle

formula_definition_triangle_height_text

Circle

formula_definition_circle_text

Area of a Circle

A = \pi r^2

Circle Sector

formula_definition_circle_sector_text

Area of a Circle Sector (Radians)

A = \frac{1}{2} r^2 \theta

Calculated using central angle

\theta

in radians.

Area of a Circle Sector (Degrees)

A = \frac{\theta}{360^\circ} \times \pi r^2

Calculated using central angle

\theta

in degrees.

Angle of a Circle Sector (Radians)

\theta = \frac{2A}{r^2}

Calculated using area A and radius r in radians.

Angle of a Circle Sector (Degrees)

\theta = \frac{360^\circ \times A}{\pi r^2}

Calculated using area A and radius r in degrees.

Area of a Triangle

A = \frac{b \times h}{2}

US Customary Length Conversions

1\text{ mi} = 1{,}760\text{ yd} \\ 1\text{ yd} = 3\text{ ft} \\ 1\text{ ft} = 12\text{ in}

Metric Length Conversions

1\text{ m} = 10\text{ dm} \\ 1\text{ dm} = 10\text{ cm} \\ 1\text{ cm} = 10\text{ mm}

Metric Prefixes

\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

US & Metric Length Conversions

\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

\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\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\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)

P_{\text{total}} = P_{\text{outer}} + P_{\text{inner}}

Perimeter of a Rectangle

P = 2(w + h)

Perimeter of a Square

P = 4s

Perimeter of a Right-angled Triangle

P = a + b + \sqrt{a^2 + b^2}

Perimeter of a Polygon

P = s_1 + s_2 + \dots + s_n

Perimeter of a Triangle

P = a + b + c

Circumference of a Circle

P = 2\pi r = \pi d

Where

r

is the radius,

d

is the diameter (

d = 2r

), and

\pi \approx 3.14159

Perimeter of a Circle Sector (Radians)

P = 2r + \theta r

Calculated using central angle

\theta

in radians.

Perimeter of a Circle Sector (Degrees)

P = 2r + \frac{\theta}{360^\circ} \times 2\pi r

Calculated using central angle

\theta

in degrees.

Rectangle Diagonals Law

formula_rectangle_diagonals_congruency_content

7th Grade Angle Fundamentals

Acute Angle

formula_definition_acute_angle_text

Obtuse Angle

formula_definition_obtuse_angle_text

Triangle

formula_definition_triangle_text

Sum of Angles in a Triangle

a + b + c = 180^\circ

Z-Rule (Alternate Angles)

formula_rule_z_text

F-Rule (Corresponding Angles)

formula_rule_f_text

X-Rule (Vertically Opposite Angles)

formula_rule_x_text

Linear Equations

Linear Equation (Slope-Intercept Form)

y = ax + b

The slope-intercept form of a linear equation, where a represents the slope (also called the gradient) and b represents the y-intercept.

Slope of a Line

a = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

Calculated from two points

(x_1, y_1)

and

(x_2, y_2)

on the line.

Property of Parallel Lines

formula_linear_parallel_slope_text

Functions

Coordinates of a Point

(x, y)

x: horizontal position
y: vertical position

Midpoint coordinates of a segment

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 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

d: Distance, A(x₁, y₁), B(x₂, y₂)

Continuity of a Function

formula_definition_continuity_text

Increasing & Decreasing Functions

formula_definition_increasing_decreasing_functions_text

Intervals of Increase & Decrease

formula_definition_increasing_decreasing_intervals_text

8th Grade Area & Pythagoras

Pythagorean Theorem

a^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 = \sqrt{b^2 - \left(\frac{a}{2}\right)^2}

Equilateral Triangle Law

formula_equilateral_triangle_law_content

Height of an Equilateral Triangle

h = a \frac{\sqrt{3}}{2}

Area of an Equilateral Triangle

A = a^2 \frac{\sqrt{3}}{4}

30-60-90 Triangle Law

formula_special_30_60_90_law_content

Area of a 30-60-90 Triangle

A = a^2 \frac{\sqrt{3}}{2}

where a is the shortest leg.

Ratio of Inscribed Square to Circle

\frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{2}{\pi}

Ratio of Inscribed Circle to Square

\frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi}{4}

8th Grade Angles & Radians

Convert Degrees to Radians

\text{rad} = \text{deg} \times \frac{\pi}{180^\circ}

Convert Radians to Degrees

\text{deg} = \text{rad} \times \frac{180^\circ}{\pi}

Volume & Surface Area

Box (Rectangular Prism)

formula_definition_box_text

Volume of a Box

V = l \times w \times h

Cube

formula_definition_cube_text

Volume of a Cube

V = s^3

Volume of Composite Shapes

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

V_{\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 \times w + l \times h + w \times h)

Surface Area of a Cube

A = 6s^2

Surface Area of Composite Shapes

A_{\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

\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

formula_definition_cylinder_text

Volume of a Cylinder

V = \pi r^2 h

Cylinder Sector (Wedge)

formula_definition_cylinder_sector_text

Volume of a Cylinder Sector

V = \frac{\theta}{360^\circ} \pi r^2 h

Calculated using radius

r

, angle

\theta

in degrees, and height

h

Surface Area of a Cylinder

A = 2\pi r^2 + 2\pi r h

Surface Area of a Cylinder Sector

A = 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

r

, central angle

\theta

in degrees, and height

h

Prism

formula_definition_prism_text

Volume of a General Prism

V = \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 \times \text{Base Area} + \text{Perimeter} \times \text{height}

Sphere

formula_definition_sphere_text

Volume of a Sphere

V = \frac{4}{3} \pi r^3

Spherical Shell

formula_definition_spherical_shell_text

Volume of a Spherical Shell

V = \frac{4}{3} \pi \left(R^3 - r^3\right)

Calculated using outer radius

R

and inner radius

r

Surface Area of a Sphere

A = 4 \pi r^2

Surface Area of a Spherical Shell

A = 4 \pi \left(R^2 + r^2\right)

Calculated as the sum of outer and inner surface areas using outer radius

R

and inner radius

r

Pyramid

formula_definition_pyramid_text

Right Pyramid

formula_definition_right_pyramid_text

Oblique Pyramid

formula_definition_oblique_pyramid_text

Volume of a Pyramid (Square)

V = \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

formula_definition_cone_text

Volume of a Cone

V = \frac{1}{3} \pi r^2 h

Volume of a Pyramid or Cone (General)

V = \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 = s^2 + 2 s l

Where

s

is the base side length and

l

is the slant height.

Slant Height of a Pyramid

l = \sqrt{h^2 + \left(\frac{s}{2}\right)^2}

Where

h

is the perpendicular height and

s

is the base side length.

Surface Area of a Cone

A = \pi r^2 + \pi r l

Where

r

is the base radius and

l

is the slant height.

Slant Height of a Cone

l = \sqrt{h^2 + r^2}

Where

r

is the base radius and

h

is the perpendicular height.

Volume Unit Conversion Rule

\text{if}\ 1\ \text{unit} = k\ \text{sub-units} \implies 1\ \text{unit}^3 = k^3\ \text{sub-units}^3

Metric Volume Conversions

1\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\text{ yd}^3 = 27\text{ ft}^3 \\ 1\text{ ft}^3 = 1{,}728\text{ in}^3

Scaling Dimensions Rules

formula_scaling_dimensions_rules_content