Asked by Fred
The total surface area of a cone is the sum of the area of the base and the lateral surface area. The formula is πr² + πrl, where r is the radius and l is the slant height
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Answered by
GPT-5 mini
AI
That is correct.
Explanation and useful forms
- Base area = πr².
- Lateral surface area = πrl. (If you "unwrap" the lateral surface it becomes a circular sector of radius l whose arc length is the base circumference 2πr; sector area = ½·(arc length)·(radius) = ½·(2πr)·l = πrl.)
- Total surface area = πr² + πrl = πr(r + l).
If you know the vertical height h instead of slant height l, use l = sqrt(r² + h²) (by the Pythagorean theorem) and substitute to get TSA = πr² + πr√(r² + h²).
Units: surface area is in square units (e.g., cm²). If the cone has no base (open-top conical surface), use only the lateral term πrl.
Explanation and useful forms
- Base area = πr².
- Lateral surface area = πrl. (If you "unwrap" the lateral surface it becomes a circular sector of radius l whose arc length is the base circumference 2πr; sector area = ½·(arc length)·(radius) = ½·(2πr)·l = πrl.)
- Total surface area = πr² + πrl = πr(r + l).
If you know the vertical height h instead of slant height l, use l = sqrt(r² + h²) (by the Pythagorean theorem) and substitute to get TSA = πr² + πr√(r² + h²).
Units: surface area is in square units (e.g., cm²). If the cone has no base (open-top conical surface), use only the lateral term πrl.
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