Cones Calculator Index

Comprehensive collection of calculators for cones and conical shapes in 3D geometry

Basic Cone Forms

Cone (V = ⅓πr²h)

Circular cone - rotation solid from triangle around an axis

Elliptic Cone a ≠ b

Cone with elliptical base - two different semi-axes

Truncated Cones (Frustums)

Truncated Cone Frustum

Cut cone between two parallel circles

Truncated Elliptic Cone

Cut elliptic cone between two ellipses

Double Cones (Bicones)

Bicone Double

Two cones with common base - mirror symmetric

Double Cone

Variant of double cone with specific geometry

Special Cone Forms

Pointed Pillar

Cylinder with conical top - combination of two basic forms

Rounded Cone Smooth

Cone with rounded tip - smooth transitions

About Cones and Conical Solids

Cones are fundamental rotation solids with diverse applications in mathematics, engineering, and nature:

Engineering - Funnels, nozzles, valves
Architecture - Roofs, towers, domes
Nature - Volcanoes, cones, shells

Mathematics - Conic sections, geometry
Industry - Conveyors, containers
Traffic - Pylons, warning signals

Fundamental Cone Formulas

Circular Cone

Volume: V = ⅓πr²h
Surface: M = πrs

Truncated Cone

V = ⅓πh(R² + Rr + r²)
Frustum formula

Elliptic Cone

Volume: V = ⅓πabh
Elliptical base

Slant Height

s = √(r² + h²)
Pythagorean in cone

Conic Sections: Plane cuts through a cone create the classical curves: circle, ellipse, parabola, and hyperbola - foundations of analytical geometry.

Practical Applications

Engineering & Industry

Funnels and nozzles: Optimal material flow
Valves: Precise flow control
Cyclones: Particle separation by rotation

Architecture & Construction

Towers: Gothic spires, church towers
Roofs: Conical roofing
Foundations: Truncated cone foundations

Traffic & Safety

Traffic cones: Visibility and stability
Warning signals: Aerodynamic shape
Road marking: Temporary barriers

Science & Nature

Volcanology: Conical volcano structures
Optics: Light cones and ray geometry
Acoustics: Sound propagation in cones

Quick Reference

⅓πr²h

Cone Volume

πrs

Surface Area

√(r² + h²)

Slant Height

⅓πabh

Elliptic

⅓πh(R² + Rr + r²)

Truncated Cone

Historical Context

Apollonius of Perga (ca. 200 BC): Systematic study of conic sections in his work "Conics".

Archimedes (287-212 BC): Calculated cone volume using the method of exhaustion.

Modern Application: Cone shapes in aerospace, fluid dynamics, and architecture.

Properties

🔺 Rotation solid: Rotation of line around axis

📐 Conic sections: Circle, ellipse, parabola, hyperbola

⚡ Flow: Optimal shape for fluids

🏗️ Statics: Good pressure distribution in structures

🎯 Focusing: Concentration of rays