Overview
This lecture covers how to calculate the volume of various 3D shapes, including cylinders, spheres, cones, rectangular prisms, and cubes, using formulas and example problems.
Volume of a Cylinder
- Volume of a cylinder = π × r² × h.
- Example: radius = 4 inches, height = 6 inches.
- 4² = 16; 16 × 6 = 96; Volume = 96π cubic inches.
- Using π ≈ 3.14, decimal volume ≈ 301.44 cubic inches.
Volume of a Sphere (Given Surface Area)
- Sphere volume formula = (4/3)πr³.
- Sphere surface area = 4πr².
- Example: surface area = 256π ft².
- 256π = 4πr² ⇒ r² = 64 ⇒ r = 8 ft.
- Volume = (4/3)π(8³) = (2048/3)π ≈ 2144.7 cubic feet.
Volume of a Cone
- Cone volume formula = (1/3)πr²h.
- Example: radius = 5 cm, height = 8 cm.
- 5² = 25; 25 × 8 = 200; Volume = (200/3)π cubic centimeters.
- Decimal approximation ≈ 209.44 cubic centimeters.
Volume of a Rectangular Prism
- Prism volume = length × width × height.
- Example: 4 ft × 5 ft × 6 ft = 120 cubic feet.
Volume of a Cube (Given Diagonal)
- Cube volume = x³, where x = side length.
- Longest diagonal of cube = x√3.
- Example: diagonal = 12.1243 cm.
- x = 12.1243 ÷ √3 ≈ 7 cm.
- Volume = 7³ = 343 cubic centimeters.
Key Terms & Definitions
- Volume — Amount of space inside an object, measured in cubic units.
- Radius (r) — Distance from the center to the edge of a circle or sphere.
- Height (h) — Vertical distance from base to top of a 3D object.
- Surface Area — Total area covering the surface of a 3D shape.
- Rectangular Prism — A box-shaped figure with 6 rectangular faces.
- Cube — A special prism where all sides are equal length.
Action Items / Next Steps
- Practice calculating volume for different shapes using the given formulas.
- Review key volume and surface area formulas for common 3D shapes.