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Cone Volume Calculator

A right circular cone is a three-dimensional geometric shape that tapers smoothly from a flat circular base to a point called the apex or vertex. This

Mathematics
Formula & Methodology
Cone Volume = (1 ÷ 3) × π × Radius² × Height = 1/3 πr²h Slant Height: s = √(Radius² + Height²) = √(r² + h²) Total Surface Area: SA = πr(r + s) = πr² + πrs

To calculate cone volume: 1. Measure the circular base radius (r) and vertical perpendicular height (h) from the base center to the apex. 2. Calculate the base area: πr². 3. Multiply base area by height and divide by 3: Volume = (1/3) × π × r² × h. The volume of a cone is always exactly one-third of a cylinder with the same radius and height.

Cone Volume ((1/3)πr²h)

150.7964

Overview

A right circular cone is a three-dimensional geometric shape that tapers smoothly from a flat circular base to a point called the apex or vertex. This free Cone Volume Calculator computes the exact cubic capacity, slant height, and surface area of any cone.

Cone volume calculations are essential in agriculture (estimating stockpiles of grain, sand, or gravel), geological volcanic magma flow models, funnel manufacturing, and culinary baking molds.

Worked Examples

Example 1: Cone with Radius = 4, Height = 9

Inputs: Radius = 4, Height = 9

  1. Base Area: π × 4² = 16π ≈ 50.27.
  2. Volume: (1/3) × 50.27 × 9 = 50.27 × 3 = 150.80 cu units.
  3. Slant height: √(4² + 9²) = √(16 + 81) = √97 ≈ 9.85.
Final Answer: Volume: 150.80, Slant Height: 9.85

Example 2: Gravel Stockpile (Radius 12 ft, Height 8 ft)

Inputs: Radius = 12, Height = 8

  1. Volume: (1/3) × π × 12² × 8 = (1/3) × π × 144 × 8 = 384π ≈ 1,206.37 cu ft.
  2. Convert to cubic yards: 1,206.37 ÷ 27 ≈ 44.68 cu yds.
Final Answer: 1,206.37 cu ft (44.68 cu yds)

Example 3: Ice Cream Cone (Radius 1.5 in, Height 5 in)

Inputs: Radius = 1.5, Height = 5

  1. Volume = (1/3) × π × 1.5² × 5 = (1/3) × π × 2.25 × 5 ≈ 11.78 cu in.
Final Answer: 11.78 cu in

How to Use This Calculator

  1. Enter the base radius of the cone into the first field.
  2. Enter the vertical perpendicular height into the second field.
  3. Click "Calculate" to determine the volume.
  4. Review the slant height and surface area below.
  5. Click "Reset" to analyze another cone.

Reference Table

Radius (r)Height (h)Slant Height (s)Volume (1/3 πr²h)
345.0037.70
499.85150.80
51213.00314.16
61011.66376.99
101518.031570.80

Frequently Asked Questions

What is slant height?
Slant height (s) is the distance from the outer edge of the circular base to the apex along the outside sloping surface, calculated as s = √(r² + h²).
Why is a cone's volume 1/3 of a cylinder?
By Cavalieri's principle and calculus integration, any right cone or pyramid has a volume equal to exactly one-third of the prism or cylinder that bounds it.
How do you calculate the volume of a truncated cone (conical frustum)?
A truncated cone with bottom radius R and top radius r has volume V = (1/3)πh(R² + r² + R·r).
What is the lateral area of a cone?
The lateral area (the curved side wall without the base) is calculated as Lateral Area = π × r × s, where s is the slant height.
How do you find height if volume and radius are known?
Rearrange the formula: Height = (3 × Volume) ÷ (π × Radius²).

Related Calculators

🧮Cylinder Volume Calculator🧮Volume Calculator🧮Sphere Volume Calculator
Reviewed & Authored by: Dr. Elizabeth Vance, PhD Applied Mathematics · Last Updated: 2026-10-06
Computed via conical circular integration and Pythagorean slant height derivation.