Lesson 12.2: Surface Area of Prisms and Cylinders

Introduction

• Focus on surface area of prisms and cylinders.
• Importance of understanding new vocabulary and terms.

Prisms

Definition of a Prism

• A polyhedron composed entirely of polygons (no curved parts).
• Features two congruent, parallel faces known as the bases.
• Examples include rectangular prisms, cubes, pentagonal prisms, and others.
  • Bases can be any of the congruent, parallel faces (e.g., top/bottom, front/back).
  • Right prisms have perpendicular bases; oblique prisms have slant heights.

Important Concepts

• Surface Area (SA): The sum of the areas of all faces of a prism.
  • SA = 2B + Ph
  • B: Area of the base.
  • P: Perimeter of the base.
  • H: Height of the prism.
  • Lateral Area (LA): Sum of areas of the lateral faces (excluding bases).

Cylinders

Definition of a Cylinder

• A solid with congruent circular bases in parallel planes (not a polyhedron due to curved edges).

Surface Area Formula

• Similar to prisms but adapted to circular bases.
• SA = 2B + Ph translates to:
  • SA = 2πr² + 2πrh
  • r: Radius of the circular base.
  • h: Height of the cylinder.
  • Full formula accounts for both the circular bases and the lateral surface area.

Example Problems

Example 1: Rectangular Prism

• Given: Dimensions of a rectangular prism (e.g., 16 cm x 4 cm x 9 cm).
• Solution: Calculate SA using the formula.
  • Identify bases, calculate B and P.
  • Use SA = 2B + Ph to find total surface area.

Example 2: Triangular Prism

• Given: Equilateral triangular prism with side length and height.
• Solution: Use specific formula for triangular base area: (s²√3)/4.
  • Compute B and perimeter, then SA.
  • Exact and approximate answers provided.

Example 3: Cylinder

• Given: Radius and height.
• Solution: Apply SA formula for cylinders.
  • Use values for r and h to compute exact and approximate SA.

Example 4: Unknown Height of Cylinder

• Given: Diameter and total surface area.
• Solution: Find radius, use formula to solve for unknown height.
  • Keep calculations in exact form as long as possible for better accuracy.

Nets

• Definition: 2D representation of a 3D shape, showing all faces unfolded.
• Can visualize how shapes like prisms and cylinders form when folded.
• Useful for understanding surface area in a visual manner.
• Example: Net of a cube or hexagonal prism.

Conclusion

• Emphasis on understanding and applying formulas for surface area of prisms and cylinders.
• Importance of exact calculations and using calculator functions (e.g., π button for accuracy).