Truncated Cone Surface Area Calculator (Frustum)

A truncated cone (conical frustum) has three surfaces — two parallel circular ends and a slanted lateral surface.

SA = π × R² + π × r² + π × (R + r) × l

Where R is the larger radius, r is the smaller radius, h is the vertical height between the two circles, and l is the slant height:

l = √((R − r)² + h²)

The l accounts for the taper — it’s longer than h whenever R ≠ r.

Worked example — lampshade fabric: A drum lampshade tapers from 14" wide at the bottom to 10" wide at the top, 12" tall. R = 7", r = 5", h = 12". Slant: l = √(4 + 144) = √148 ≈ 12.17 in. Lateral surface (just the fabric, no top/bottom rings): π × (7 + 5) × 12.17 ≈ 458.6 sq in = 3.18 sq ft.

A roll of lampshade fabric is typically 30" wide × 1 yd long = 30 × 36 = 1,080 sq in. One yard covers two shades like this with room for waste. Actual lampshades need 15-20% extra for the curved seam allowance.

Worked example — galvanized steel bucket: A 12-quart bucket tapers from R = 7" (top) to r = 5" (bottom), 11" tall. l = √(4 + 121) = √125 ≈ 11.18 in. Lateral surface: π × 12 × 11.18 ≈ 421.4 sq in. Bottom disc: π × 25 ≈ 78.5 sq in. No top (open bucket). Total sheet metal needed: 500 sq in = 3.47 sq ft per bucket (plus seam allowance, hardware, handle).

Where truncated cone surface matters in practice:

• Lampshades. Fabric, paper, parchment for shade materials.
• Buckets and pails. Sheet metal for galvanized or stainless steel buckets.
• Plastic pots. Tapered planter pots — injection-molded surface area.
• Hopper tank exteriors. Conical hoppers on industrial silos.
• Champagne flutes. Glass material for tapered drinking vessels.
• Conical heat shields. Aerospace re-entry vehicles use this shape.
• Beverage cups. Coffee shop hot cups taper from top rim to base.

The slant height formula details:

The slant l is NOT the same as h. For a perfectly cylindrical shape (R = r), l = h. For a truncated cone (R > r), the slant is longer than h because it includes the radial drop (R − r) and the vertical drop (h).

Visualize: imagine cutting the slant with a string from rim to rim. The string goes (R − r) inward and h downward — Pythagoras gives the slant.

Open vs. closed truncated cones:

• Closed (both ends): total π(R² + r²) + π(R+r)l.
• Open top (bucket, pot, lampshade): π × r² + π(R+r)l (one circular end + slant).
• Open both ends (truncated cone tube): just π(R+r)l.

Pick the variant that matches your fabrication: lampshades and open buckets are open-top, gym medicine balls and capsule end caps are closed.

Sanity check:

• R = r (no taper): SA = 2πR² + 2πRh. Matches cylinder formula. ✓
• r = 0 (full cone): SA = πR² + πR × √(R² + h²) = πR² + πR × slant. Matches cone formula. ✓