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Area of a Rectangle

A = l \times w

Area of a rectangle.

Area of a Triangle

A = \frac{1}{2} b h

Area of a triangle from base and height.

Area of a Trapezoid

A = \frac{1}{2}(a+b)h

Area of a trapezoid with parallel sides a and b.

Area of a Parallelogram

A = b h

Area of a parallelogram.

Area of an Ellipse

A = \pi a b

Area enclosed by an ellipse.

Surface Area of a Sphere

A = 4\pi r^2

The total surface area of a sphere.

Surface Area of a Cylinder

A = 2\pi r(r + h)

Total surface area of a closed cylinder.

Heron's Formula

A = \sqrt{s(s-a)(s-b)(s-c)}

Area of a triangle from its three sides, s = (a+b+c)/2.

Area of a Sector

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

Area of a circular sector (θ in radians).

Area of a Square

A = a^2

Space enclosed by a square.

Area of a Rhombus

A = \frac{1}{2} d_1 d_2

Area from the two diagonals of a rhombus.

Area of a Kite

A = \frac{1}{2} d_1 d_2

Area from the two diagonals of a kite.

Area of a Regular Polygon

A = \frac{1}{2} a P

Area from apothem and perimeter.

Surface Area of a Cube

A = 6a^2

Total surface area of a cube.

Surface Area of a Cuboid

A = 2(lw + lh + wh)

Total surface area of a rectangular box.

Surface Area of a Cone

A = \pi r(r + l)

Total surface area of a cone with slant height l.

Curved Surface of a Cylinder

A = 2\pi r h

Side (curved) surface area of a cylinder.

Area of Regular Polygon

A = \frac{1}{2} a p

Area from apothem and perimeter.

Physics Laws & TheoremsAdvanced

Sector Area (Radians)

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

Area of a circular sector with angle in radians.

TrigonometryIntermediate

Triangle Area (SAS)

\text{Area} = \tfrac{1}{2}ab\sin C

Area from two sides and the included angle.

TrigonometryIntermediate

Heron's Formula

\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}

Triangle area from the three side lengths, where s is the semi-perimeter.

CalculusAdvanced

Surface of Revolution

S = 2\pi\int_a^b f(x)\sqrt{1+(f'(x))^2}\,dx

Surface area generated by revolving a curve about the x-axis.

CalculusIntermediate

Definite Integral (Area)

\int_a^b f(x)\,dx = \text{signed area under } f

The net area between a curve and the x-axis from a to b.

PhysicsBasic

Pressure

P = \frac{F}{A}

Force per unit area.

Kepler's Second Law

\frac{dA}{dt} = \text{const}

A line from a planet to the Sun sweeps equal areas in equal times.

Electricity & MagnetismAdvanced

Parallel-Plate Capacitance

C = \frac{\varepsilon_0 A}{d}

Capacitance from plate area and separation.

Fluid MechanicsBasic

Pressure

P = \frac{F}{A}

Force per unit area.

Electronics & CircuitsIntermediate

Wire Resistance

R = \frac{\rho L}{A}

Resistance from resistivity, length and area.

Biology & Life SciencesBasic

Surface Area to Volume

SA:V = \frac{\text{surface area}}{\text{volume}}

Ratio governing cellular exchange efficiency.

Biology & Life SciencesIntermediate

Surface-to-Volume Ratio

SA:V = \frac{\text{surface area}}{\text{volume}}

Ratio governing cell exchange efficiency.

Health & FitnessIntermediate

Body Surface Area (Mosteller)

BSA = \sqrt{\frac{h \times m}{3600}}

Total skin surface area of the body.

EngineeringBasic

Stress

\sigma = \frac{F}{A}

Internal force per unit area in a material.

EngineeringAdvanced

Second Moment of Area (Rect.)

I = \frac{bh^3}{12}

Resistance of a rectangular section to bending.

Materials & CivilBasic

Stress

\sigma = \frac{F}{A}

Internal force per unit cross-sectional area.

Materials & CivilAdvanced

Rectangular Area Moment of Inertia

I = \frac{b h^3}{12}

Second moment of area for a rectangular cross-section.

Materials & CivilIntermediate

Hydraulic Radius

R_h = \frac{A}{P}

Cross-sectional flow area divided by wetted perimeter.

Aviation & AeronauticsIntermediate

Lift Equation

L = \tfrac{1}{2}\rho v^2 S C_L

Aerodynamic lift produced by a wing depends on air density, speed squared, area and lift coefficient.

Aviation & AeronauticsAdvanced

Aspect Ratio

AR = \dfrac{b^2}{S}

Ratio of wingspan squared to wing area; higher AR reduces induced drag.

Aviation & AeronauticsIntermediate

Wing Loading

WL = \frac{W}{S}

Aircraft weight per unit wing area.

Everyday MathBasic

Paint Needed

\text{Gallons} = \frac{A}{C}

Amount of paint required for a wall area.

Everyday MathBasic

Tiles Needed

N = \frac{A_{floor}}{A_{tile}}

Number of tiles to cover a floor.

Everyday MathBasic

Room Area

A = l \times w

Floor area of a rectangular room.

Geography & EarthBasic

Population Density

D = \frac{P}{A}

Number of people per unit area.

Square Meters to Square Feet

\text{ft}^2 = \text{m}^2 \times 10.7639

Convert area from square meters to square feet.

Acres to Square Meters

\text{m}^2 = \text{acre} \times 4046.86

Convert acres to square meters.

Hectares to Acres

\text{acre} = \text{ha} \times 2.47105

Convert hectares to acres.

Square Km to Square Miles

\text{mi}^2 = \text{km}^2 \times 0.386102

Convert area from square kilometers to square miles.