Library / Foundations: Maths / FMA-58
Foundations: Maths
Total surface area of a cone
Whole outside area of a closed cone: the curved surface πrℓ plus the circular base πr².
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- FMA-58
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- v1
- Field
- Foundations: Maths
- Result
- A [m²]
A closed cone has a curved surface and a circular base. The curved surface unrolls into part of a circle and has area πrℓ, where ℓ = √(r² + h²) is the slant height along the surface from the rim to the tip. The base adds πr². For the curved area only, omit the base term. Lengths must be in one unit.
Whole outside area of a closed cone: the curved surface πrℓ plus the circular base πr². The slant height ℓ comes from the radius and the perpendicular height.
Calculator
- Slant height ℓ = √(r² + h²)5 m
- Curved area only, πrℓ47.12 m²
- Base area, πr²28.27 m²
- Volume, πr²h/337.7 m³
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A cone of base radius 3 m and height 4 m has slant height 5 m (the 3–4–5 triangle), so its curved surface is 47.12 m² and its total surface 75.4 m².
- The slant height, not the vertical height, is the right length in the curved-surface formula: using h instead of ℓ would understate the area by 20 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| r Base radius | 0.1 to 50 m | 1.288 | 15,730 | 15,730 | +10% | +16.7 % |
| h Perpendicular height | 0.1 to 100 m | 56.56 | 971.2 | 914.6 | +10% | +4.07 % |
Sweep of Base radius
| Base radius (m) | Total surface area (m²) |
|---|---|
| 0.1 | 1.288 |
| 6.337 | 275.4 |
| 12.57 | 1,018 |
| 18.81 | 2,249 |
| 25.05 | 3,968 |
| 31.29 | 6,176 |
| 37.52 | 8,873 |
| 43.76 | 12,060 |
| 50 | 15,730 |
Sweep one variable
Variables and how each one contributes
Total surface area
What the equation gives you. Everything else feeds it.
Base radius
The radius of the circular base, in metres. Raises the area faster than in proportion, because the radius appears in the slant height and in both area terms. Raising it by 0.5 m (to 3.5 m) moves the result by +28.6 %, from 75.4 to 96.93 m².
Perpendicular height
The perpendicular height from the base to the tip, in metres. Raises the area through the slant height; the effect is smaller for a tall cone than a flat one. Raising it by 0.5 m (to 4.5 m) moves the result by +5.1 %, from 75.4 to 79.25 m².
Worked example with the values above
- Slant height: ℓ = √(3² + 4²) = 5 m.
- Curved surface πrℓ = π × 3 × 5 = 47.12 m², and base πr² = 28.27 m².
- A = 75.4 m².
Assumptions and limits
- A right circular cone with a flat circular base.
- Radius and height in the same length unit; area in that unit squared.
Common mistakes
- Using the vertical height instead of the slant height in the curved-surface term.
- Forgetting the base area for a closed cone.
Statistics and uncertainty
Area scales with the square of a length, so a 1 % error in the radius gives about a 1.5 to 2 % error in the area. Measure the slant height directly if the cone is hard to measure by height.
Where it is used
- Surface area of a nose cone, a funnel or a conical pile.
- Material for a conical cover.
Maths you need
- Pythagoras: the square of the hypotenuse is the sum of squares
- Circle area: πr²