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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.

Law of Sines

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Relates sides to opposite angles in any triangle.

Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

Generalizes Pythagoras to any triangle.

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.

Sine Double Angle

\sin(2\theta) = 2\sin\theta\cos\theta

Double-angle identity for sine.

Cosine Double Angle

\cos(2\theta) = \cos^2\theta - \sin^2\theta

Double-angle identity for cosine.

Sine Angle Sum

\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B

Angle addition formula for sine.

Cosine Angle Sum

\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B

Angle addition formula for cosine.

Perimeter of a Rectangle

P = 2(l + w)

Total length around a rectangle.

Perimeter of a Triangle

P = a + b + c

Sum of the three sides of a triangle.

Diagonal of a Rectangle

d = \sqrt{l^2 + w^2}

Length of the diagonal of a rectangle.

Sum of Interior Angles

S = (n - 2)\times 180^\circ

Total of interior angles of an n-sided polygon.

Interior Angle (Regular Polygon)

\theta = \frac{(n-2)\times 180^\circ}{n}

Measure of each interior angle of a regular polygon.

Tangent Double Angle

\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}

Double-angle identity for tangent.

Interior Angle Sum

S = (n-2)\times 180\degree

Sum of interior angles of an n-sided polygon.

Each Interior Angle (Regular)

\theta = \frac{(n-2)\times 180\degree}{n}

Measure of one interior angle of a regular polygon.

Exterior Angle (Regular)

\theta = \frac{360\degree}{n}

Measure of one exterior angle of a regular polygon.

Circumradius

R = \frac{abc}{4K}

Radius of a triangle circumscribed circle.

Inradius

r = \frac{K}{s}

Radius of a triangle inscribed circle.

Tangent Half-Angle

\tan\frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta}

Expresses the tangent of a half angle.

Cofunction Identity

\sin\theta = \cos(90\degree - \theta)

Sine and cosine of complementary angles are equal.

Arc Length (Radians)

s = r\theta

Arc length from radius and central angle in radians.

Sector Area (Radians)

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

Area of a circular sector with angle in radians.

Sine Ratio (SOH)

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

Ratio of the side opposite an angle to the hypotenuse.

Cosine Ratio (CAH)

\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

Ratio of the side adjacent to an angle to the hypotenuse.

Sine of a Sum

\sin(A+B) = \sin A\cos B + \cos A\sin B

Angle addition formula for sine.

Sine of a Difference

\sin(A-B) = \sin A\cos B - \cos A\sin B

Angle subtraction formula for sine.

Cosine of a Sum

\cos(A+B) = \cos A\cos B - \sin A\sin B

Angle addition formula for cosine.

Cosine of a Difference

\cos(A-B) = \cos A\cos B + \sin A\sin B

Angle subtraction formula for cosine.

Tangent of a Sum

\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}

Angle addition formula for tangent.

Tangent of a Difference

\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A\tan B}

Angle subtraction formula for tangent.

Double Angle: Sine

\sin 2\theta = 2\sin\theta\cos\theta

Sine of twice an angle.

Double Angle: Cosine

\cos 2\theta = \cos^2\theta - \sin^2\theta

Cosine of twice an angle (also 2cos²θ−1 or 1−2sin²θ).

Double Angle: Tangent

\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}

Tangent of twice an angle.

Half Angle: Sine

\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}}

Sine of half an angle.

Half Angle: Cosine

\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}

Cosine of half an angle.

Half Angle: Tangent

\tan\frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta}

Tangent of half an angle.

Law of Sines

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Relates sides of any triangle to the sines of opposite angles.

Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

Generalises Pythagoras to any triangle.

Law of Tangents

\frac{a-b}{a+b} = \frac{\tan\tfrac{A-B}{2}}{\tan\tfrac{A+B}{2}}

Relates two sides and their opposite angles.

Triangle Area (SAS)

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

Area from two sides and the included angle.

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.

Degrees to Radians

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

Convert an angle from degrees to radians.