BYTETOOLS

Cone Calculator

Solve a right circular cone from any two of radius, height and slant height, with volume, lateral and total surface area, apex angle and a cutting template.

37.6991
Volume (cm³)
5
Slant height (cm)
47.1239
Lateral area (cm²)
75.3982
Total area (cm²)

Where each number came from

  • Radius r = 3 cm
  • Vertical height h = 4 cm
  • Slant height l = √(r² + h²) = 5 cm
  • Volume = πr²h/3 = 37.6991 cm³
  • Lateral surface = πrl = 47.1239 cm²
  • Base area = πr² = 28.2743 cm²
  • Total surface = πr(r + l) = 75.3982 cm²
  • Base circumference = 2πr = 18.8496 cm
  • Half-apex angle = arcsin(r/l) = 36.87°
  • Full apex angle = 73.74°

Flat template for cutting

  • Sector radius = slant height = 5 cm
  • Sector angle = 360 × r / l = 216°
  • Arc length = base circumference = 18.8496 cm

Cut this sector from flat stock, roll it into a cone and the curved edge becomes the rim. The pattern is the finished size — add your own seam or glue allowance.

What is the Cone Calculator?

The ByteTools Cone Calculator takes any two of the three measurements that define a right circular cone — base radius, vertical height and slant height — and works out the third with l = √(r² + h²).

  • Solves from radius and height, radius and slant, or height and slant
  • Volume, lateral area, base area, total area and base circumference
  • Apex and half-apex angles from arcsin(r/l)
  • Flat cutting template with sector radius, included angle and arc length, drawn to scale
  • Radius or diameter input, in millimetres, centimetres, metres, inches or feet
  • Runs entirely in your browser — nothing is uploaded

How to use the Cone Calculator

  1. 1

    Choose which two measurements you know from the What do you know? dropdown.

  2. 2

    Say whether you are entering the base as a radius or a diameter, and pick your unit.

  3. 3

    Type the two known figures into the numbered fields.

  4. 4

    Read the volume, slant height and surface areas in the summary tiles.

  5. 5

    Open the flat template panel for the sector radius and angle, then click Copy results.

About the Cone Calculator

The ByteTools Cone Calculator takes any two of the three measurements that define a right circular cone — base radius, vertical height and slant height — and works out the third with l = √(r² + h²). From there it gives you the volume πr²h/3, the lateral surface πrl, the total surface πr(r + l), the base circumference and both the half-apex and full apex angles.

It also draws the flat pattern. Unrolled, the curved side of a cone is a circular sector with radius equal to the slant height and an included angle of 360 × r ÷ l degrees, which is exactly what you need to mark out sheet metal, card or fabric before rolling it up. The sector is drawn to scale alongside a side view of the finished cone.

Everything is computed in your browser with JavaScript, so nothing you type is uploaded or stored anywhere. That makes the calculator instant, private and usable offline once the page has loaded — handy in a workshop with patchy signal. Radii or diameters can be entered in millimetres, centimetres, metres, inches or feet, and every answer is labelled with the unit you chose.

Frequently asked questions

How do you find the slant height of a cone?

The slant height is the hypotenuse of a right triangle whose legs are the radius and the vertical height, so l = √(r² + h²). A cone with a 3 cm radius and a 4 cm height has a slant height of exactly 5 cm. The calculator can also run this backwards to find the radius or height from a known slant.

What is the formula for the volume of a cone?

Volume = πr²h ÷ 3, which is exactly one third of the cylinder that would enclose it. For a 3 cm radius and 4 cm height that gives 37.70 cm³. Note that the vertical height goes into this formula, not the slant height — using the slant is the single most common mistake.

How do I make a flat template for a cone?

Draw a circular sector with radius equal to the cone's slant height and an angle of 360 × r ÷ l degrees. For a 3 cm radius and 5 cm slant, that is a sector of radius 5 cm and angle 216°. Cut it out, roll it up and the straight edges meet to form the cone.

What is the difference between lateral and total surface area?

Lateral surface area is just the curved side, πrl, which is what you need when you are making an open cone such as a funnel or a party hat. Total surface area adds the circular base, giving πr(r + l). The calculator shows both so you can pick the right one.

Can I enter the diameter instead of the radius?

Yes. Switch the Enter the base as dropdown to Diameter and the tool halves it internally before solving. Every result is still reported in terms of the radius, which is what the standard formulas use.

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