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Cone Surface Area Calculator

Find the surface area of a cone.

Result

282.74 units²
204.20 units²
13.00

How it is calculated

A cone has a circular base and a curved side. The slant height is found from the right triangle formed by radius and height: s = √(r² + h²). The curved (lateral) area is L = π·r·s, and the total surface area adds the base: A = π·r² + π·r·s = π·r·(r + s). Radius r is the base radius and h is the straight height from base to tip, not the slant. Both must be above zero and use the same length unit; areas come out in that unit squared (shown as units²), rounded to two decimals.

Worked example

Take r = 5 and h = 12. Slant height: √(25 + 144) = √169 = 13. Lateral area: π × 5 × 13 = 204.20. Base area: π × 25 = 78.54. Total: 204.20 + 78.54 = 282.74 units². If the units are centimetres, the cone needs about 282.74 cm² of material to cover its whole surface, base included.

Try this example →

Common mistakes

  • Entering the slant height in the height field. The height must be perpendicular to the base; the slant is calculated for you.
  • Using the diameter instead of the radius, which makes the area four times too large.
  • Forgetting the base when the question asks for total surface area. Lateral area is only the curved part.
  • Mixing units, such as radius in centimetres and height in metres.
  • Using this for an open cone or a cut-off cone (frustum); the formula covers a complete, closed cone only.

About this calculator

This calculator finds a cone's slant height using the Pythagorean relation l = √(r² + h²), then computes lateral surface area as π·r·l and total surface area as π·r·(r + l) — the lateral term unrolls the cone's curved side into a flat circular sector, while adding π·r² accounts for the circular base.

Geometry students solving surface-area problems, engineers and sheet-metal fabricators calculating material needed for conical parts like funnels, chutes, or roof caps, and hobbyists figuring out paper or fabric needed for a cone-shaped craft (party hats, lampshades) use this. Enter the radius and height to get the slant height, lateral area, and total surface area.

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