4MA1 | 4.103D Volumes & Surface Area
The Fabricator: Cones & Spheres
Interactive Exploration of Volume & Surface Area Formulas
Phase 1: Simple Cone (The Anchor)
A cone is a 3D shape with a circular base and a point at the top. Let's start with easy numbers to build confidence.
The Visual Toy
Cone with radius 3 cm, height 4 cm (easy Pythagorean triple!)
Formulas
Volume of Cone:
V = (1/3)πr²h
= (1/3)π(3)²(4) = 12π ≈ 37.7 cm³
Curved Surface Area:
CSA = πrl
= π(3)(5) = 15π ≈ 47.1 cm²
Base Area:
BA = πr²
= π(3)² = 9π ≈ 28.3 cm²
✨ KEY INSIGHT:
The slant height (l = 5) forms the hypotenuse of a right triangle inside the cone. We'll use this to derive formulas!
📋 Summary: Key Formulas
CONE (Use Pythagoras: l² = h² + r²)
V = (1/3)πr²h
CSA = πrl
BA = πr²
Total SA = πrl + πr²
SPHERE & HEMISPHERE
V(sphere) = (4/3)πr³
SA(sphere) = 4πr²
V(hemi) = (2/3)πr³
SA(hemi) = 3πr² ← TRAP!