Volume of a Cone Calculator

Calculate the volume and surface area of a cone: V = ⅓πr²h. Find slant height, lateral surface area, and total surface area. Works for right circular cones.

📐 Geometry📐 V = ⅓πr²h🔢 Math
Radius r
Height h
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Volume of a Cone Calculator—V = ⅓πr²hcubic units

Step-by-Step Examples

Example 1
Classic

r=3, h=5.

  • V=⅓×π×9×5=15π≈47.12
  • l=√(9+25)=√34≈5.83
  • Lateral SA=π×3×5.83≈54.96
✓ V ≈ 47.12 cubic units
Example 2
Ice Cream Cone

r=2.5cm, h=10cm.

  • V=⅓π(6.25)(10)=20.83π≈65.45 cm³
✓ V ≈ 65.45 cm³
Example 3
Cone vs Cylinder

Same r=3, h=5: Cylinder V=πr²h=45π. Cone=15π.

  • Cone = ⅓ of cylinder with same r,h
✓ Cone = exactly ⅓ of cylinder

Real-World Applications

🍦
Food
Ice cream cones, waffle cones — volume determines serving size.
🏗️
Engineering
Funnels, hoppers, silos with conical bottoms.
📐
Geometry
Cone is a fundamental solid of revolution — cross-section of a cone gives conic sections.
🧪
Chemistry
Erlenmeyer flask approximates a cone — important for volume calculations.

Common Mistakes to Avoid

⚠️
V = ⅓ not ½

Cone volume is ⅓πr²h, exactly one-third of the cylinder with same r and h.

⚠️
Slant height ≠ height

Height h is the perpendicular from apex to base. Slant height l = √(r²+h²). These are different measurements.

⚠️
Radius not diameter

r in the formula is radius (half the base diameter). If given diameter, divide by 2 first.

Frequently Asked Questions

Why does ⅓ appear? ▾
A cone fills exactly ⅓ of its circumscribed cylinder. This can be proved by calculus (integration) or by Cavalieri's principle.
What is the slant height? ▾
The distance from the apex to any point on the base circumference: l = √(r²+h²). Used in lateral surface area = πrl.
What is the lateral surface area? ▾
The curved surface of the cone (not including the base): A_lateral = πrl where l is slant height.
What are conic sections? ▾
Slicing a cone at different angles gives: circle (horizontal), ellipse (tilted), parabola (parallel to side), hyperbola (steeper than side).
What is a frustum? ▾
A cone with the top cut off by a plane parallel to the base. Volume of frustum = ⅓πh(r₁²+r₁r₂+r₂²).

Related Math Calculators

📐 Volume📐 Pyramid Volume📐 Circle📐 Surface Area📐 Diagonal

Formula Explorer connections

Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.

Volume of a Pyramid Calculator →2D Transformation Matrix Calculator →3D Transformation Matrix Calculator →Math Formula Explorer →