Cone Calculator Online
Calculate cone volume, slant height, lateral area, and total surface area with step-by-step formulas.
Input Parameters
Live Result
Details
Steps
Calculate Slant Height using Pythagorean theorem: s = √(r² + h²)
Compute Circular Base Area: A_base = π · r²
Compute Lateral Surface Area: A_lateral = π · r · s
Compute Cone Volume: V = (1/3) · A_base · h
Calculate Total Surface Area: A_total = A_base + A_lateral
Quick Answer & Understanding Your Result
A right circular cone is a 3D geometric shape that tapers smoothly from a flat circular base to an apex point, with volume V = (1/3) · π · r² · h and slant height s = √(r² + h²).
Key Mathematical Formulas
- Volume: One-third of circumscribed cylinder: V = (1/3) · π · r² · h.
- Slant Height (s): Hypotenuse from base perimeter to apex: s = √(r² + h²).
- Lateral Surface Area (A_l): Unrolled sector mantle: A_l = π · r · s.
- Total Surface Area (A_t): Lateral mantle plus circular base: A_t = πr(r + s).
- Apex Opening Angle: Vertex divergence angle: α = 2 · arcsin(r / s).
Architectural & Manufacturing Uses
- Funnel & Hopper Fabrication: Sizing conical funnels, aggregate discharge hoppers, and grain silos.
- Traffic & Safety Equipment: Designing standard roadway warning cones and delineators.
- Aerospace & Nose Cones: Engineering supersonic projectile and rocket fairing nose cones for optimal aerodynamics.
Enter the base radius and vertical height into the calculator above to get instant volume, slant height, and surface area values.
Formula & Calculation Logic
Mathematical Formular is the radius of the circular base, h is the vertical perpendicular height, and s is the slant height calculated via the Pythagorean theorem. The unrolled lateral surface forms a circular sector with radius s.
Step-by-Step Guide
Enter Base Radius
Input the radius of the cone's circular flat base.
Enter Vertical Height
Input the perpendicular distance from the center of the base to the apex point.
Select Measurement Unit
Choose meters, centimeters, millimeters, inches, or feet.
Review Complete Metrics
Analyze the computed volume, slant height, lateral area, total area, and capacity in liters.
Worked Examples Table
| Example | Input Values | Calculated Result | Explanation |
|---|---|---|---|
| Pythagorean Cone: r = 3m, h = 4m | Slant s = √(9 + 16) = 5 m, V = (1/3)·π·9·4 = 12π ≈ 37.6991 m³, Lateral = π·3·5 = 15π ≈ 47.1239 m². | ||
| Traffic Cone: r = 15cm, h = 60cm | Slant s = √(225 + 3600) ≈ 61.85 cm, V = (1/3)·π·225·60 = 4500π ≈ 14,137.2 cm³ (14.14 Liters). |
Frequently Asked Questions
Evidence-Based Methodology & Authoritative References
Wolfram MathWorld - Cone
Euclid's Elements - Book XII (Volume of Cones and Pyramids)
Standard: Volume, Slant Height & Surface Area
Formula: Volume: V = (1/3) · π · r² · h
Last Reviewed: August 21, 2026
Primary Source: Wolfram MathWorld - Cone
Reference Document: Euclid's Elements - Book XII (Volume of Cones and Pyramids)
Computation Privacy: 100% client-side 0ms computation, zero data retention
Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.
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Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.