Geometry & Shapes

Annulus

A = π(R² − r²)

Area of an annular ring between two concentric circles.

Capsule

V = πr²h + (4/3)πr³

Volume and surface area of a capsule (cylinder + two hemispherical caps).

Circle Equation

A = πr² · C = 2πr

Area, circumference, diameter, and radius of a circle.

Circle Arc Length

s = r · θ (θ in radians)

Arc length, chord length, and sagitta from radius and central angle.

Circle Sector

A = ½ · r² · θ

Pie-slice area, arc length, and chord of a circular sector.

Circle Segment

A = ½r²(θ − sinθ)

Area, chord, height, and arc of the region between a chord and an arc.

Cone

V = ⅓πr²h · S = πr(r + ℓ)

Volume, lateral and total surface area, and slant height of a right cone.

Cone Frustum

V = ⅓πh(R² + Rr + r²)

Volume and surface area of a cone with the top sliced off — truncated cone.

Cube

V = a³ · S = 6a²

Volume, surface area, and diagonal of a cube from edge length.

Cylinder

V = πr²h · S = 2πr(r + h)

Volume, lateral and total surface area of a right circular cylinder.

Distance

d = √((x₂−x₁)² + (y₂−y₁)²)

Distance between two points in the coordinate plane (2D and 3D).

Dodecahedron

V = (15 + 7√5)/4 · a³

Volume, surface area, and circumradius of a regular twelve-faced Platonic solid.

Ellipse

A = π · a · b

Area, circumference (Ramanujan), and foci of an ellipse from semi-axes.

Ellipsoid

V = (4/3)π · a · b · c

Volume and approximate surface area of a triaxial ellipsoid.

Equilateral Triangle

A = (√3 / 4) · a²

Area, perimeter, height, and inscribed/circumscribed radii of an equilateral triangle.

Geometric Formulas

Reference shape library

Quick reference for area, perimeter, surface area, and volume of standard shapes.

Hemisphere

V = (2/3)πr³ · S = 3πr²

Volume and total surface area of a hemisphere (half a sphere).

Hexagon

A = (3√3 / 2) · a²

Area, perimeter, and inscribed/circumscribed radii of a regular hexagon.

Hexagonal Prism

V = (3√3 / 2) · a² · h

Volume and surface area of a regular hexagonal prism.

Hollow Cylinder

V = π · (R² − r²) · h

Volume and surface area of a hollow cylinder (cylindrical pipe).

Icosahedron

V = (5(3 + √5)/12) · a³

Volume, surface area, and circumradius of a regular twenty-faced Platonic solid.

Isosceles Triangle

A = ½ · b · h

Area, perimeter, height, and inscribed/circumscribed radii of an isosceles triangle.

Kite

A = ½ · d₁ · d₂

Area, perimeter, and diagonals of a kite quadrilateral.

Octagon

A = 2(1 + √2) · a²

Area, perimeter, and inscribed/circumscribed radii of a regular octagon.

Octagonal Prism

V = 2(1 + √2) · a² · h

Volume and surface area of a regular octagonal prism.

Octahedron

V = (√2 / 3) · a³

Volume, surface area, and circumradius of a regular eight-faced Platonic solid.

Parallelogram

A = b · h

Area, perimeter, and diagonals of a parallelogram.

Pentagon

A = (5/4) · a² · cot(π/5)

Area, perimeter, and inscribed/circumscribed radii of a regular pentagon.

Pentagonal Prism

V = (5/4) · a² · cot(π/5) · h

Volume and surface area of a regular pentagonal prism.

Pyramid

V = (1/3) · A_base · h

Volume and surface area of a pyramid with any regular polygon base.

Pythagorean Theorem

a² + b² = c²

Solve for any side of a right triangle given the other two.

Rectangle

A = L · W · P = 2(L + W)

Area, perimeter, and diagonal of a rectangle.

Rectangular Prism

V = L · W · H

Volume, surface area, and diagonal of a rectangular box (cuboid).

Rhombus

A = ½ · d₁ · d₂

Area, perimeter, and diagonals of a rhombus.

Right Triangle

a² + b² = c² · A = ½ab

Sides, area, perimeter, altitudes, medians, and inscribed/circumscribed radii of a right triangle.

Sphere

V = (4/3)πr³ · S = 4πr²

Volume and surface area of a sphere from radius or diameter.

Spherical Cap

V = (πh² / 3)(3r − h)

Volume and surface area of a sphere sliced by a single plane.

Spherical Segment

V = (πh / 6)(3a² + 3b² + h²)

Volume and surface area of the region between two parallel planes cutting a sphere.

Square

A = a² · P = 4a · d = a√2

Area, perimeter, and diagonal of a square from side length.

Square Pyramid Frustum

V = (h/3)(a² + a · b + b²)

Volume and surface area of a square pyramid with the top sliced off.

Tetrahedron

V = a³ / (6√2)

Volume, surface area, and circumradius of a regular four-faced Platonic solid.

Torus

V = 2π² · R · r²

Volume and surface area of a torus (doughnut) from inner and outer radii.

Trapezoid

A = ½ · (a + b) · h

Area, perimeter, and midsegment of a trapezoid.

Triangle Equations

Law of sines / cosines + Heron's

Sides, angles, area, altitudes, medians, and bisectors of any triangle (scalene, isosceles, equilateral).

Triangular Prism

V = (1/2) · b · h · L

Volume and surface area of a triangular prism.

Geometry calculators for every standard 2D shape and 3D solid: triangles (right, equilateral, isosceles, scalene), circles and circle parts (arc, sector, segment, annulus), polygons (hexagon, pentagon, octagon, parallelogram, rhombus, kite, trapezoid), and solids (cube, sphere, cylinder, cone, pyramid, prism, torus, the five Platonic solids).

Each calculator solves for any variable in the shape's equations — give it the inputs you know, get the rest. Inline diagrams label every variable.

When to use these calculators

Use these for math homework verification, engineering quick-checks, woodworking and construction layout, and 3D-printing volume estimation. The triangle family (right triangle, Pythagorean theorem) and circle family are the most-used. The 3D solids handle surface area and volume for cube, cylinder, sphere, cone, and 14 other solids.

Many calcs include inline labeled SVG diagrams so the formula variables match the shape you're solving on the page.

Frequently Asked Questions

How does the Pythagorean theorem calculator work?: Given any two sides of a right triangle (the two legs, or one leg and the hypotenuse), the Pythagorean Theorem calculator solves for the missing side using a² + b² = c². It also handles the inverse: given hypotenuse and one leg, solve for the other leg.
What's the difference between Circle, Circle Sector, Circle Segment, and Annulus?: Circle = full disk (area, circumference). Circle Sector = pie slice between two radii. Circle Segment = region bounded by a chord and an arc. Annulus = ring between two concentric circles. Each has its own area / arc-length / chord formulas.
Are surface area and volume both supported for 3D solids?: Yes — every 3D solid calculator (sphere, cylinder, cone, cube, pyramid, prism, torus, dodecahedron, icosahedron, octahedron, tetrahedron, hemisphere, spherical cap, spherical segment, cone frustum, square pyramid frustum, hexagonal/octagonal/pentagonal/rectangular/triangular prism) computes both surface area and volume from the shape's defining dimensions.