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VolumeCalcKit
Geometry

Cone Volume Calculator

Calculate cone volume, slant height, base area, and lateral surface area from radius or diameter and vertical height.

Instant results
Real-time calculation
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Standard SI formulas
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rhslant (s)
Technical Dimension Schematic100% Vector
Dimension Input Mode
Cone Geometryrhslant (s)

V = ⅓πr²h • s = √(r² + h²)

Cone Calculation Results
Decimals:
Volume in M3Live Output
3.1416m3
Additional Computed Dimensions
Slant Height (s)
3.1623 m
Base Surface Area
3.1416 m²
Total Surface Area
13.0762 m²
Equivalent Standard Volume Units
Liters (L)
3,141.5927 l
Cubic Centimeters (cm³)
3,141,592.6536 cm3
Cubic Feet (ft³ / cu ft)
110.9443 ft3
Cubic Inches (in³)
191,711.7461 in3
Cubic Yards (yd³)
4.109 yd3
US Liquid Gallons (gal)
829.921 us_gal
Imperial Gallons (UK gal)
691.0538 imp_gal

Mathematical Formula & Step-by-Step Derivation

SI Standard Math
General Formula:
V = ⅓ × π × r² × h = ⅓ × π × (d / 2)² × h
Live Calculation With Your Input Values:
11. Cone Volume Formula
Formula: V = ⅓ × π × r² × h
Substituted: V = ⅓ × π × (1 m)² × (3 m)
Step Result: V = 3.1416 m3

Normalized in SI: ⅓ × π × (1 m)² × (3 m) = 3.1416 m³

22. Slant Height
Formula: s = √(r² + h²)
Substituted: s = √(1² + 3²)
Step Result: s = 3.1623 m (3.1623 m)
33. Total Surface Area
Formula: A = πr(r + s)
Substituted: A = π × 1 × (1 + 3.1623)
Step Result: A = 13.0762 m²
Cone Mathematical Formulas & Geometry Relationships
Volume (Radius & Height):V = ⅓ × π × r² × h
Slant Height (Pythagorean):s = √(r² + h²)
Lateral Surface Area:A_lat = π × r × s
Total Surface Area:A_tot = πr(r + s)
Worked Engineering Example: Grain Hopper / Silo Cone

Problem: Determine the capacity of a conical funnel hopper with radius 1.5 m and height 3.0 m.

• Base Area A = π × (1.5)² = 7.0686 m²

• Volume V = ⅓ × 7.0686 × 3.0 = 7.0686 m³

• Liquid Capacity: 7.0686 m³ × 1,000 L/m³ = 7,068.58 Liters (1,867.33 US Gallons)

• Slant Height s = √(1.5² + 3.0²) = √(2.25 + 9.0) = √11.25 ≈ 3.3541 meters

• Lateral Steel Area = π × 1.5 × 3.3541 ≈ 15.806 m²

Common Use Cases
  • Angle of Repose in Bulk Storage: In agriculture and mining, free-flowing aggregates (gravel, sand, grain) dumped from a conveyor form natural conical piles whose slope angle equals the material's friction angle of repose (typically 30° to 45°).
  • Cone vs. Cylinder Ratio: Exactly 3 cones of equal radius and height fit into one cylinder of the same dimensions (Ratio is 1:3).
  • Frustum of a Cone (Truncated Cone): If the top tip of the cone is cut off by a plane parallel to the base, the volume of the remaining frustum with radii R and r is V = ⅓πh(R² + Rr + r²).

Complete Guide: How to Calculate the Volume of a Right Circular Cone

A right circular cone is a three-dimensional geometric solid that tapers smoothly from a circular base to an apex point directly perpendicular to the base center. The volume of any cone is precisely one-third (⅓) the volume of a cylinder having the identical base radius and vertical height.

  1. Measure the base radius (r): Measure the circular bottom from center to edge. If measuring the full diameter, divide by 2 (r = d / 2).
  2. Measure vertical height (h): Measure the perpendicular altitude from the center of the base directly to the apex. Do not measure along the slanted side.
  3. Compute the circular base area: Square the radius and multiply by Pi (A_base = π × r²).
  4. Multiply by height and divide by 3: Multiply the base area by vertical height and divide by 3: V = ⅓ × π × r² × h.
  5. Calculate slant height (s) if needed: Using the Pythagorean theorem, the diagonal side distance is s = √(r² + h²).

Frequently Asked Questions

How do you calculate the volume of a cone?
Use the formula V = (1/3) × π × r² × h, where r is the base radius and h is the vertical perpendicular height. Multiply Pi by the radius squared, multiply by height, and divide by 3.
What is the difference between cone vertical height and slant height?
Vertical height (h) is the perpendicular straight line distance from the center of the base to the apex. Slant height (s) is the distance along the outer angled surface from the outer edge of the base to the apex. They form a right triangle where s² = r² + h².
Can I calculate the volume of a cone if I only have the diameter?
Yes. Divide the diameter by 2 to get the radius (r = d / 2), then apply V = (1/3) × π × r² × h. Alternatively, write the formula directly in terms of diameter: V = (π / 12) × d² × h ≈ 0.2618 × d² × h.
What is the formula for the volume of a conical pile of gravel or sand?
Because gravel or sand dumps into a circular cone, measure the diameter across the base circle and the vertical height at the peak. The volume is V = (1/3) × π × (d/2)² × h. For example, a 6m wide, 2m high pile has volume V = (1/3) × π × (3)² × 2 = 18.85 m³.
How many liters of water can a conical tank hold?
One cubic meter of volume equals 1,000 liters. Once you calculate the cone volume in cubic meters using V = (1/3) × π × r² × h, multiply by 1,000 to get total capacity in liters.
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