Surface Area of a Cone
Total surface area of a cone with slant height l.
Formula
Variables
Example
Radius 3, slant 5: A = π·3·(3+5) ≈ 75.4 square units.
How it works
This formula gives the total outside area of a right circular cone: the flat circular base plus the curved side. The curved part is not a circle, but when it is unwrapped it forms a sector of a circle, and that geometry leads to the term \u03c0rl; adding the base area \u03c0r\u00b2 gives \u03c0r(r + l). The slant height is used instead of the vertical height because it is the length along the side that determines the curved surface area.
When to use it
Use this formula when you need the amount of material needed to cover a cone, such as for a paper cone, roof, funnel, or ice-cream cone. It is also useful in geometry problems that ask for the total surface area of a cone from its radius and slant height.
Worked example, step by step
Radius r = 4 cm, slant height l = 7 cm A = \u03c0r(r + l) A = \u03c0 \u00d7 4 \u00d7 (4 + 7) A = \u03c0 \u00d7 4 \u00d7 11 A = 44\u03c0 A \u2248 138.2 cm\u00b2
Common mistakes to avoid
- Using the vertical height instead of the slant height in the formula
- Forgetting to include the base area when the question asks for total surface area
- Calculating with the diameter instead of the radius
Did You Know?
An ice-cream cone's paper wrap, unrolled, is a circular sector — the source of the πrl term.
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