Cone Calculator

Calculate the volume and surface area of a right circular cone from its radius or diameter and perpendicular height. Check the slant height, convert units and follow the calculation steps.

Calculate Your Cone

Enter radius or diameter, then add the height. The connected radius and diameter fields describe the same base.

Distance from the center of the circular base to its edge.

Distance across the base through its center. Diameter = 2 × radius.

Straight distance from the tip to the base center, not along the sloping side.

h r ℓ
h = perpendicular height · ℓ = slant height. Illustration is not drawn to scale.

You can use different input units. Changing an input unit converts its measurement while preserving the cone’s size.

Comparing shapes? A cone has one-third the volume of a cylinder with the same base radius and height. Explore the cylinder volume calculator to compare their enclosed space.

QUICK START

How to Use the Cone Calculator

Start with the circular base and perpendicular height. The calculator finds the slant height, volume and surface areas from these measurements.

  1. Enter radius or diameter

    Use the base radius if you measured from its center to the edge, or diameter if you measured across the center. Enter either value; the connected field updates automatically.

  2. Add the perpendicular height

    Enter the straight distance from the cone’s tip to the center of its base. This is the height inside the cone, not the sloping distance along its side. Choose the matching unit for each input.

  3. Choose result units and rounding

    Select the units for your results and two, four or six decimal places. Lengths appear in ordinary units, areas in square units and volume in cubic units. Input measurements can use different units.

  4. Review the areas and calculation steps

    Check volume, slant height and the separate base and curved areas. Total surface area includes both the curved side and circular base. Select Copy Result to save the measurements, or Clear to start again.

Check the shape first: This tool assumes a complete right circular cone with its tip directly above the base center. A cone with its top cut off is a frustum and needs a different calculation.

WORKED EXAMPLES

Cone Calculations Step by Step

Calculate volume from radius or diameter, then use the slant height to find surface area. Decimal results below are rounded to four places.

EXAMPLE 01 · VOLUME

Calculate volume from radius and height

A cone has a radius of 5 centimeters and a perpendicular height of 12 centimeters. Multiply the circular base area by the height, then divide by three.

GIVEN MEASUREMENTS
Radius (r)
5 cm
Perpendicular height (h)
12 cm
  1. Find the circular base area

    B = π × r²

    B = π × 5² = 25π cm²

  2. Multiply by height and divide by three

    V = B × h ÷ 3

    V = 25π × 12 ÷ 3 = 100π cm³

  3. Round the final answer

    V ≈ 314.1593 cm³

RESULT Volume ≈ 314.1593 cm³

Use the perpendicular height of 12 cm in the volume formula, rather than the slant height.

EXAMPLE 02 · SURFACE AREA

Find total surface area from a diameter

A cone has a base diameter of 6 inches and a perpendicular height of 4 inches. Find the radius and slant height before adding the curved area to the base area.

GIVEN MEASUREMENTS
Diameter (d)
6 in
Perpendicular height (h)
4 in
  1. Convert diameter to radius

    r = d ÷ 2 = 6 ÷ 2 = 3 in

  2. Calculate the slant height

    ℓ = √(r² + h²)

    ℓ = √(3² + 4²) = 5 in

  3. Add the base and curved surface areas

    S = πr² + πrℓ

    S = 9π + 15π = 24π in²

RESULT Total surface area ≈ 75.3982 in²

Base area ≈ 28.2743 in². Curved surface area ≈ 47.1239 in².

Choose the area that matches your task

To cover only the sloping side, use the curved surface area πrℓ. To include the circular base as well, use total surface area πr² + πrℓ. Enter your cone measurements to see both results separately.

UNDERSTAND THE FORMULAS

Cone Formulas at a Glance

Perpendicular height determines volume, while slant height determines the curved surface area. The diagrams below show how these measurements fit together.

h r
Volume = base area × height ÷ 3 V = πr²h ÷ 3. Use the height perpendicular to the base, rather than the length of the sloping side.
ℓ Curved side r Base
Total surface area = curved side + base S = πrℓ + πr². When flattened, the curved side forms a circular sector whose radius is the slant height ℓ.
r = base radius, d = diameter, h = perpendicular height, and ℓ = slant height. Formulas apply to a right circular cone; diagrams are schematic and not drawn to scale.
Measurement Formula Units
Radius from diameter r = d ÷ 2 Length, e.g. cm
Slant height ℓ = √(r² + h²) Length, e.g. cm
Base area B = πr² Square, e.g. cm²
Volume from radius V = πr²h ÷ 3 Cubic, e.g. cm³
Volume from diameter V = πd²h ÷ 12 Cubic, e.g. cm³
Curved surface area A = πrℓ Square, e.g. cm²
Total surface area S = πr(r + ℓ) Square, e.g. cm²

Why are height and slant height different?

A vertical cross-section through the center contains a right triangle with sides r, h and ℓ. Slant height is its hypotenuse, so ℓ = √(r² + h²). For a cone with a positive radius, the slant height is longer than the perpendicular height.

Why does the diameter formula divide by twelve?

Replacing r with d ÷ 2 makes r² equal to d² ÷ 4. The volume formula also divides by three. Combining these factors gives V = πd²h ÷ 12. Entering a diameter as a radius would make the calculated volume four times too large.

Keep units consistent before substituting: Radius and height must use the same length unit in a manual calculation. The calculator converts mixed input units automatically. Round the final answer rather than rounding the slant height or π early.

COMMON QUESTIONS

Frequently Asked Questions

Answers about cone height, surface areas, mixed units and the shapes supported by this calculator.

Can I calculate a cone using its diameter?

Yes. Enter the diameter and perpendicular height. The radius field updates using r = d ÷ 2. The calculator then finds volume and surface areas. The equivalent volume formula using diameter is V = πd²h ÷ 12.

What is the difference between height and slant height?

Perpendicular height runs straight from the tip to the base center. Slant height runs along the side from the tip to the base edge. For a right circular cone, ℓ = √(r² + h²). Use perpendicular height for volume and slant height for curved surface area. The cone reference on Wolfram MathWorld explains these measurements and their formulas.

What if I know the slant height instead of the height?

First find perpendicular height with h = √(ℓ² − r²), then enter that value in the height field. Radius and slant height must use the same units in this calculation. For a cone with positive height, the slant height must be greater than the radius.

Does total surface area include the base?

Yes. Total surface area adds the circular base πr² to the curved side πrℓ. If you need only the sloping exterior, use the separate curved surface area result. It excludes the base.

Can radius and height use different units?

Yes. Choose the correct unit beside each measurement. The calculator converts them to consistent units before calculating. For example, you can enter radius in centimeters and height in meters. The result selector controls the displayed length, square and cubic units.

Can I calculate a cone with its top cut off?

A cone cut parallel to its base forms a conical frustum. This tool calculates a complete cone, so its results should not be used directly for a frustum. Frustum volume is V = πh(R² + Rr + r²) ÷ 3, where R and r are the two circular radii and h is the perpendicular distance between their planes.

Does this calculator support an oblique cone?

The calculator assumes a right circular cone with its tip directly above the base center. An oblique circular cone still has volume πr²h ÷ 3 when h is the perpendicular height. However, this tool’s slant height and surface area results do not apply to that shape.

Can I use the volume to estimate container capacity?

For a container whose interior forms a complete right circular cone, use its internal radius and internal perpendicular height. Divide a volume in cm³ by 1,000 to convert it to liters. Outside measurements include wall thickness and can overestimate capacity. A container with a flat, truncated tip needs a frustum calculation instead.

READY TO CALCULATE?

Find Your Cone’s Volume and Surface Area

Enter the base radius or diameter and perpendicular height. Check the enclosed volume, slant height and surface areas with clear calculation steps.

Use Cone Calculator