Cone Surface Area Calculator

A cone has two surfaces — the lateral (slanted side) surface and the circular base.

SA = π × r² + π × r × l

Where r is the base radius and l is the slant height. The slant height l is computed from the radius and the perpendicular height h:

l = √(r² + h²)

For example, a cone with r = 3, h = 4 has slant l = 5 (the classic 3-4-5 right triangle).

Worked example — party hat pattern: A child’s birthday party hat: base radius r = 4 in (just fits a kid’s head), height h = 8 in. Slant height: l = √(16 + 64) = √80 ≈ 8.94 in.

You want the FABRIC area for the hat — that’s the lateral surface only, no base (the head goes there). Lateral surface: π × 4 × 8.94 ≈ 112.4 sq in.

But you cut this from a flat sheet of paper as a SECTOR, not as a curved 3D shape. The sector has:

• Radius = slant height (l = 8.94 in)
• Arc length = base circumference (2πr = 25.13 in)
• Sector angle = (arc / full-circle) × 360° = (25.13 / (2π × 8.94)) × 360° ≈ 161°

So you’d cut a 161° sector of an 8.94" radius circle, roll it into a cone, and tape the edges.

Worked example — traffic cone material: An 18" tall orange cone with 10" base diameter. r = 5", h = 18", l = √(25 + 324) = √349 ≈ 18.68". Lateral surface = π × 5 × 18.68 ≈ 293 sq in. Add 60% for the welded seam and base ring: ~470 sq in of plastic per cone.

Where cone surface area matters:

• Party hat patterns. Card stock or fabric template — flat sector shape.
• Ice cream cone wrappers. Paper labels around a sugar or waffle cone.
• Megaphones. Sheet-metal lateral area for a hand-held megaphone.
• Funnel making. Sheet metal or glass for laboratory or kitchen funnels.
• Speaker drivers. Cone-shaped paper or polypropylene speaker diaphragms.
• Traffic cones and witch hats. Plastic injection molding, calculated by surface for material cost.
• Lampshade tops. Conical lampshades that taper to a point.

The slant height vs. perpendicular height confusion:

These are different. Slant height is along the cone’s surface; perpendicular height is straight up from the base center.

For ALL surface area calculations, use slant height (l). For volume calculations, use perpendicular height (h).

Mistake-prone formula: people see “πr × h” and assume that’s lateral surface. It’s not — that would be the area of a cylinder’s curved side, not a cone’s. A cone has πr × l.

Lateral only vs. total surface:

• Open cone (no base): funnels, megaphones, party hats — lateral only, πrl.
• Closed cone (with base): traffic cones with bottom plate — total πr² + πrl.

Sanity check:

• h = 0: cone collapses to a flat disc of radius r. SA = πr² (just the base). ✓
• Slant length l = r: cone is so flat it’s almost a disc.
• l = r√2 (45° angle): a standard “pizza party hat” shape.