Geometry Formulas

Distance Formula

Reference for 2D distance d = √((x₂-x₁)²+(y₂-y₁)²) and 3D extension. Covers midpoint formula, Manhattan distance, and coordinate geometry applications.

Golden Ratio Formula

The golden ratio φ = (1 + √5) / 2 ≈ 1.618 appears in geometry, art, architecture, and nature. Learn the formula, Fibonacci connection, and design applications.

Midpoint Formula

Reference for midpoint formula M = ((x1+x2)/2, (y1+y2)/2) with 2D and 3D examples. Covers distance formula and uses in geometry, navigation, and graphics.

Area of a Regular Polygon

Calculate the area of any regular polygon using A = (n*s^2*cot(pi/n))/4. Works for pentagons, hexagons, octagons, and any n-sided polygon with equal sides.

Brahmagupta's Formula

Calculate the area of a cyclic quadrilateral using Brahmagupta's formula: A = sqrt((s-a)(s-b)(s-c)(s-d)). Generalizes Heron's formula.

Ellipsoid Volume Formula

Calculate the volume of an ellipsoid from its three semi-axis lengths. Used in geology, medicine, and physics applications.

Euler's Polyhedron Formula

Euler's Polyhedron Formula V - E + F = 2 relates vertices, edges, and faces of convex polyhedra. Examples and proof explanation.

Shoelace Formula

Calculate polygon area from vertex coordinates using the Shoelace Formula. Works for any simple polygon with known vertices.

Spherical Cap Volume

Calculate spherical cap volume V = πh²(3R−h)/3 and surface area. Used in engineering, fluid fill calculations, and 3D modeling for dome and bowl-shaped solids.

2D Rotation Matrix

The 2D rotation matrix transforms coordinates by angle θ, rotating points around the origin using cosine and sine functions.

Circle Formulas

Reference for circle formulas: area A = πr², circumference C = 2πr, arc length, sector area, and chord length. Covers unit circle with worked examples.

Conic Sections Formulas

Standard equations for ellipses, parabolas, and hyperbolas. Learn conic section formulas with key properties and worked examples.

Heron's Formula

Reference for Heron's formula for triangle area from three side lengths: A = sqrt(s(s-a)(s-b)(s-c)). Includes the semi-perimeter and step-by-step examples.

Parametric Equations

Parametric equations define curves using x(t) and y(t) as functions of a parameter t, enabling complex curves like circles and cycloids.

Pythagorean Theorem

Pythagorean theorem: a² + b² = c² for right triangles. Covers finding hypotenuse, missing legs, Pythagorean triples, and distance formula applications.

Spherical Geometry Formulas

Great circle distance and spherical excess formulas for navigation and geodesy. Calculate distances on a sphere with worked examples.

Triangle Formulas

Reference for key triangle formulas: base-height area, Heron's formula, law of sines, law of cosines, and angle sum rules with worked examples.

Volume Formulas

Reference for volume formulas for cubes, prisms, cylinders, spheres, cones, and pyramids. Includes the formula and a worked example for each shape.

Volume of a Frustum

Calculate the volume of a frustum (truncated cone or pyramid). Learn the formula with step-by-step examples for both shapes.

Volume of a Torus

Calculate the volume and surface area of a torus from major and minor radii. Covers the Pappus centroid theorem approach with step-by-step worked examples.

Arc Length Formula

Reference for arc length L = rθ (radians) and L = (θ/360) × 2πr (degrees). Covers track design, belt drives, and gear pitch with radian/degree conversion.

Diagonal of a Rectangle

Calculate the diagonal length of a rectangle using the Pythagorean theorem. Used in construction, screens, and layout design.

Sector Area Formula

Reference for sector area A = (θ/360) × πr² in degrees and A = ½r²θ in radians. Covers arc length L = rθ with examples for pie charts and sprinkler coverage.

Surface Area of a Cylinder

Calculate the total surface area of a cylinder including the top, bottom, and curved side. Used in manufacturing and packaging.

Volume of a Torus

Calculate volume V = 2π²Rr² and surface area of a torus (donut shape). Used in engineering tolerances, 3D modeling, and manufacturing ring-shaped components.

Area of a Circle

Calculate the area of a circle using A = πr². Learn how to find the area from the radius or diameter with step-by-step examples.

Area of a Parallelogram

Reference for parallelogram area formulas A = base × height and A = ab sin(θ). Covers perimeter (2a+2b), rectangles, rhombuses, and worked examples.

Area of a Rectangle

Reference for rectangle area A = l × w and perimeter P = 2(l+w). Covers square as special case, diagonal, and applications for flooring and land area.

Area of a Trapezoid

Reference for the trapezoid area formula A = half times (a plus b) times h. Calculate area from two parallel sides and height with step-by-step examples.

Area of a Triangle

Reference for triangle area formulas: A=½bh, Heron's formula from three sides, coordinate shoelace formula, and SAS with sine. Step-by-step worked examples.

Area of an Ellipse

Reference for ellipse area A = π × a × b from semi-major and semi-minor axes, plus perimeter approximation using the Ramanujan formula and eccentricity.

Circumference of a Circle

Calculate the circumference of a circle using C = 2πr or C = πd. Find the distance around any circle with step-by-step examples.

Surface Area of a Sphere

Calculate the surface area of a sphere using SA = 4πr². Find the outer surface area of balls, globes, and spherical objects.

Volume of a Cone

Calculate the volume of a cone using V = (1/3)πr²h. A cone holds exactly one-third the volume of a cylinder with the same base and height.

Volume of a Cylinder

Calculate cylinder volume using V = πr²h and lateral surface area using 2πrh. Includes worked examples for tanks, pipes, cans, and cylindrical containers.

Volume of a Pyramid

Calculate the volume of a pyramid using V = (1/3)Bh. Works for any pyramid shape — square, rectangular, or triangular base.

Volume of a Sphere

Calculate the volume of a sphere using V = (4/3)πr³. Find how much space a ball or globe occupies with step-by-step examples.