Surface Area of Cones
A cone is a “circular pyramid.” Its surface area = circular base + curved lateral area.
Introduction
Past Knowledge
Circle area (11.2.3). Pyramid SA (12.2.2). Pythagorean theorem.
Today's Goal
Derive and apply SA = πr² + πrℓ for cones.
Future Success
Sphere SA (12.2.4), volume of cones (12.3.2).
Key Concepts
Cone Surface Area
= base, = lateral area. Slant height:
Connection to Pyramid Formula
Pyramid: SA = B + ½Pℓ. For cone: B = πr², P = 2πr → ½(2πr)ℓ = πrℓ. Same formula, circular base!
Worked Examples
Given r and ℓ
r = 3, ℓ = 5
SA = 24π ≈ 75.4
Find ℓ First
r = 5, h = 12
SA = 90π ≈ 282.7
Common Pitfalls
Using h Instead of ℓ
SA uses slant height ℓ. Volume uses perpendicular height h. If only h is given, find ℓ via Pythagorean.
Real-Life Applications
Ice Cream Cones
The waffle cone material = lateral area πrℓ. If you want to coat it in chocolate, you need the lateral surface area.
Traffic Cones
The reflective material on a traffic cone = lateral area. Manufacturers calculate πrℓ for material ordering.
Practice Quiz
Loading...