Cone Volume & Surface Area Calculator

A right circular cone is a geometric solid with a circular base and a vertex that rises perpendicularly above the center of the base. This calculator computes both the volume and total surface area from the radius and height.

Cone formulas

For a cone with radius r, height h and slant height l:

• Volume: V = (1/3) × π × r² × h
• Total surface area: SA = π × r × (r + l)
• Slant height: l = √(r² + h²)

The slant height is found using the Pythagorean theorem, as it forms a right triangle with the radius and the height.

Example 1: cone with integer dimensions

Problem: A cone has radius r = 6 cm and height h = 8 cm.

1. Slant height:
   • l = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
2. Volume:
   • V = (1/3) × π × 6² × 8 = (1/3) × π × 36 × 8 ≈ 301.59 cm³.
3. Surface area:
   • SA = π × 6 × (6 + 10) = π × 6 × 16 ≈ 301.59 cm².

Answer: V ≈ 301.59 cm³, SA ≈ 301.59 cm², l = 10 cm.

Example 2: cone with decimal measurements

Problem: A cone has radius r = 3.5 m and height h = 5.2 m.

1. Slant height:
   • l = √(3.5² + 5.2²) = √(12.25 + 27.04) = √39.29 ≈ 6.27 m.
2. Volume:
   • V = (1/3) × π × 3.5² × 5.2 ≈ 66.73 m³.
3. Surface area:
   • SA = π × 3.5 × (3.5 + 6.27) ≈ 107.43 m².

Answer: V ≈ 66.73 m³, SA ≈ 107.43 m², l ≈ 6.27 m.

Common uses of the cone calculator

• Computing the capacity of funnels, traffic cones and conical containers.
• Estimating materials for conical roofs, tents and umbrellas.
• Determining volumes of sand, gravel or bulk material piles.
• Solving solid geometry problems in mathematics courses.
• Designing decorative and architectural conical elements.
• Calculating the surface area of ice cream cones and food packaging.

Common mistakes when working with cones

• Using the vertical height instead of the slant height in the surface area formula.
• Forgetting the 1/3 factor in the volume formula.
• Confusing lateral area with total surface area (which includes the base).
• Not computing the slant height when only radius and height are known.

Pro tip

Remember that a cone's volume is exactly one-third of a cylinder's volume with the same base and height. This relationship can help you mentally verify your calculations.