Surface Area of Cones – SA = πr² + πrl
A cone's surface area is its flat circular base (πr²) plus its curved lateral surface (πrl), where l is the slant height – the distance from the apex down to the edge of the base, measured along the curved surface. Together: SA = πr² + πrl.
Unlike a cylinder's height, a cone's slant height usually isn't given directly – it has to be calculated from the radius and vertical height using the Pythagorean relationship l² = r² + h², since the radius, vertical height, and slant height form a right triangle inside the cone.
Calculating a Cone's Surface Area
SA = πr² + πrl. If only r and h are known, first find l using l = √(r² + h²).
Find the surface area of a cone with radius 4 cm and slant height 7 cm.
SA = π(4)² + π(4)(7) = 16π + 28π = 44π ≈ 138.23 cm².
A cone has radius 6 cm and height 8 cm. Find its slant height.
l = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Real-Life Application
- Party hats: the paper or card needed to make a cone-shaped hat.
- Ice cream cone wrappers: packaging designed to match a cone's curved surface.
- Traffic cone manufacturing: estimating material for cone-shaped safety markers.
Key Takeaways
- A cone's surface area formula is SA = πr² + πrl.
- Slant height l is found from radius and vertical height using l² = r² + h².
- This links a cone's surface area directly to the Pythagorean theorem.
Practice: Surface Area of Cones
Surface Area of Cones
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