Cone calculator — volume, slant height and surface area

How to use the cone calculator

1. Enter the Base radius, half the width of the circular base.
2. Enter the Vertical height, from the base straight up to the tip.
3. Read Volume (⅓πr²h) and Slant height (√(r² + h²)).
4. Read the areas: Lateral surface (πrs) for the sloping side, Base area (πr²), and Total surface (πr(r + s)). Apex half-angle gives the steepness of the side.

What it does and when to use it

Ice-cream cones, funnels, traffic cones, party hats, conical roofs and piles of sand or gravel are all close to cones. The volume tells you how much a funnel or a heap holds. The lateral surface is the paper or sheet metal needed to roll the side, which is why paper hat templates start from the slant height. The apex angle matters when cutting a funnel or matching a countersink.

For a cone with its top cut off, choose the frustum in the surface area calculator or the volume calculator. The cylinder calculator lets you compare a cone with the cylinder of the same size.

How it works

With base radius r and vertical height h:

• Slant height: s = √(r² + h²), because r, h and s form a right triangle
• Volume: V = ⅓ × π × r² × h
• Lateral surface: πrs
• Base area: πr²
• Total surface: πr² + πrs = πr(r + s)
• Apex half-angle: arctan(r ÷ h)

Worked examples

A party hat, step by step. Base radius 6 cm, height 8 cm.

1. Slant height: √(6² + 8²) = √(36 + 64) = √100 = 10.
2. Volume: ⅓ × π × 36 × 8 = 96π = 301.5928947 cm³.
3. Curved side: π × 6 × 10 = 60π = 188.4955592 cm².
4. Base: π × 36 = 113.0973355 cm². Total surface: 188.4955592 + 113.0973355 = 301.5928947 cm².

The tool shows Slant height (√(r² + h²)): 10 and Lateral surface (πrs): 188.4955592, so the paper for the hat (no base) needs about 189 cm². The apex half-angle is 36.869898°. By coincidence, the volume and total surface share the number 301.59 here, but one is in cm³ and the other in cm².

A narrow funnel. Radius 2 cm, height 9 cm. The volume is 37.69911184 cm³, the slant height 9.219544457 cm and Apex half-angle: 12.528808°, so the tip is about 25° wide in total.

Limits and tips

• Both inputs must be greater than zero.
• Only right circular cones (tip directly over the centre) are covered.
• There is no unit menu; use one unit for both inputs.
• The calculation runs in your browser.