Enter radius and vertical height to calculate the cone surface areas.
Solve from radius + vertical height or radius + slant height, then separate lateral/base material and verify the flattened sector.
Keep linear inputs in a consistent unit, verify that side lengths or coordinate points can form the stated shape, and treat sketches as explanatory unless explicitly marked to scale. Coordinate formulas can become unstable for nearly degenerate shapes, so invalid or nearly collinear inputs are rejected rather than silently returning misleading values.
Use the product first, then apply these tool-specific checks to verify assumptions, interpret the result, and hand it off safely without moving the primary workflow below generic content.
The perpendicular height runs from the apex to the center of the base, while slant height runs along the side. Surface area uses slant height; volume uses perpendicular height. If a drawing labels only h or l, confirm which segment it means before entering the number. Mixing them can produce a plausible-looking but materially wrong result.
A party hat or open funnel may need only the curved lateral surface, while a closed cone needs the base too. State the physical boundary before using a total. For material estimates, separate lateral and base areas so another person can see exactly what was included rather than receiving a single unexplained square-unit number.
For radius r and vertical height h, check l squared equals r squared plus h squared. If slant height is supplied directly, it must be at least as large as the radius. The flattened side is a circular sector whose arc length equals the base circumference, giving an independent check on any cone-net cutting plan.
The formulas are unit-agnostic only when every length uses the same unit. Convert mixed measurements before calculating. If the result will drive fabric, sheet metal, paper, or coating quantities, apply seam allowance, overlap, trimming, kerf, and waste as separate project assumptions instead of hiding them inside the geometric surface area.
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