Geometry & Trigonometry

Areas, volumes, angles and trigonometric identities

Geometry & Trigonometry brings together the essential formulas for measuring shapes, spaces, angles, and triangles, from areas and perimeters to volumes and trig identities. It’s a handy reference for students, exam revision, and anyone using maths in practical work like engineering, design, construction, or science. These formulas help you solve problems quickly, check results accurately, and understand the relationships behind everyday and advanced calculations.

Pythagorean Theorem

a^2 + b^2 = c^2

Relates the sides of a right triangle.

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.

Equation of a Circle

NEWIntermediate

(x - h)^2 + (y - k)^2 = r^2

The standard form of the equation of a circle with centre (h, k) and radius r.

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.