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Area of a Circle

A = \pi r^2

The area enclosed by a circle.

Circumference of a Circle

C = 2\pi r

The perimeter (boundary length) of a circle.

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.

Volume of a Sphere

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

The volume enclosed by a sphere.

Volume of a Cylinder

V = \pi r^2 h

The volume of a cylinder.

Volume of a Cone

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

The volume of a cone.

Volume of a Cube

V = a^3

The volume of a cube.

Volume of a Box

V = l w h

Volume of a rectangular cuboid.

Volume of a Pyramid

V = \frac{1}{3} B h

Volume of any pyramid.

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.

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.

Pythagorean Identity

\sin^2\theta + \cos^2\theta = 1

The fundamental trigonometric identity.

Tangent Identity

\tan\theta = \frac{\sin\theta}{\cos\theta}

Defines tangent in terms of sine and 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.

Arc Length

s = r\theta

Length of a circular arc (θ in radians).

Area of a Sector

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

Area of a circular sector (θ in radians).

Distance Between Two Points

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

Euclidean distance in the plane.

Midpoint Formula

M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

Midpoint of a line segment.

Slope of a Line

m = \frac{y_2 - y_1}{x_2 - x_1}

Steepness of a straight line.

Perimeter of a Square

P = 4a

Total length around a square.

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.

Perimeter of a Parallelogram

P = 2(a + b)

Total length around a parallelogram.

Perimeter of a Rhombus

P = 4a

All four sides of a rhombus are equal.

Perimeter of a Trapezoid

P = a + b + c + d

Sum of all four sides of a trapezoid.

Perimeter of a Regular Polygon

P = n s

Total length around a regular polygon.

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.

Diagonal of a Square

d = a\sqrt{2}

Length of the diagonal of a square.

Diagonal of a Rectangle

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

Length of the diagonal of a rectangle.

Space Diagonal of a Cuboid

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

Longest diagonal through a rectangular box.

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.

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.

Cotangent Identity

\cot\theta = \frac{\cos\theta}{\sin\theta}

Cotangent as the ratio of cosine to sine.

Reciprocal Trig Identities

\sec\theta = \tfrac{1}{\cos\theta}, \ \csc\theta = \tfrac{1}{\sin\theta}

Secant and cosecant as reciprocals.

Tangent Double Angle

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

Double-angle identity for tangent.

Area of Regular Polygon

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

Area from apothem and perimeter.

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.

Number of Diagonals

D = \frac{n(n-3)}{2}

Count of diagonals in an n-sided polygon.

Frustum Volume

V = \frac{1}{3}\pi h(R^2 + Rr + r^2)

Volume of a truncated cone.

Torus Volume

V = 2\pi^2 R r^2

Volume of a donut-shaped solid.

Spherical Cap Volume

V = \frac{\pi h^2}{3}(3R - h)

Volume of a slice cut from a sphere.

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.

Secant Definition

\sec\theta = \frac{1}{\cos\theta}

Reciprocal of cosine.

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.

Tangent Ratio (TOA)

\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

Ratio of the opposite side to the adjacent side.

Cosecant

\csc\theta = \frac{1}{\sin\theta}

Reciprocal of sine.

Secant

\sec\theta = \frac{1}{\cos\theta}

Reciprocal of cosine.

Cotangent

\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}

Reciprocal of tangent.

Tangent Identity

\tan\theta = \frac{\sin\theta}{\cos\theta}

Tangent equals sine divided by cosine.

Pythagorean Identity

\sin^2\theta + \cos^2\theta = 1

The fundamental trig identity from the unit circle.

Identity: 1 + tan² = sec²

1 + \tan^2\theta = \sec^2\theta

Derived Pythagorean identity dividing by cos².

Identity: 1 + cot² = csc²

1 + \cot^2\theta = \csc^2\theta

Derived Pythagorean identity dividing by sin².

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.

Product: sin A cos B

\sin A\cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)]

Product-to-sum identity.

Product: cos A cos B

\cos A\cos B = \tfrac{1}{2}[\cos(A-B) + \cos(A+B)]

Product-to-sum identity.

Product: sin A sin B

\sin A\sin B = \tfrac{1}{2}[\cos(A-B) - \cos(A+B)]

Product-to-sum identity.

Sum to Product: sin A + sin B

\sin A + \sin B = 2\sin\tfrac{A+B}{2}\cos\tfrac{A-B}{2}

Sum-to-product identity.

Sum to Product: sin A − sin B

\sin A - \sin B = 2\cos\tfrac{A+B}{2}\sin\tfrac{A-B}{2}

Sum-to-product identity.

Sum to Product: cos A + cos B

\cos A + \cos B = 2\cos\tfrac{A+B}{2}\cos\tfrac{A-B}{2}

Sum-to-product identity.

Sum to Product: cos A − cos B

\cos A - \cos B = -2\sin\tfrac{A+B}{2}\sin\tfrac{A-B}{2}

Sum-to-product identity.

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.

Radians to Degrees

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

Convert an angle from radians to degrees.

Euler's Formula

e^{i\theta} = \cos\theta + i\sin\theta

Links complex exponentials to trigonometry.

De Moivre's Theorem

(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta

Powers of complex numbers in trig form.