Lecture Notes: Calculating the Surface Area of Prisms and Cylinders

Introduction

• Surface Area Definition: Surface area is the total area of all the external surfaces of a 3D object.
  • For 2D shapes like squares, area is inside the shape, calculated as base x height.
  • For 3D objects like cubes, surface area is the sum of all outside areas.

Calculating Surface Area of Prisms

• Example 1: Rectangular Prism

  • Identify the shape of the base; in this case, rectangles.
  • Calculate the area of all faces and sum them.

• Example 2: Triangular Prism

  • Steps to Calculate Surface Area:
    1. Identify the shape of the base (triangles).
    2. Find the area of the bases:
       • Use the formula for area of a triangle: ( \frac{1}{2} \times \text{base} \times \text{height} ).
       • Height is found using Pythagorean theorem: ( a^2 + b^2 = c^2 ).
       • Example: Base = 6 inches, height found as ( \sqrt{27} ), area of one base = 15.6 inches².
    3. Double the area for two bases: 31.2 inches².
    4. Find the area of the rectangular faces:
       • Each face: base x height, e.g., 63 inches² for one.
       • Sum for all faces: 63 x 3 = 189 inches².
    5. Total surface area: 220.2 inches².

Calculating Surface Area of Cylinders

• Base Shape: Circles
• Steps to Calculate Surface Area:
  1. Area of the base using ( \pi r^2 ):
     • Diameter = 2 feet, radius = 1 foot.
     • Area of one base = 3.14 feet², both bases = 6.28 feet².
  2. Area of the curved surface (like a paper towel unrolled is a rectangle):
     • Length is the height (7 feet) and width is the circumference (2( \pi r ) = 6.28 feet).
     • Area of the rectangle = 44 feet².
  3. Total Surface Area: 50.3 feet².

Calculating Surface Area of Hexagonal Prism

• Base Shape: Hexagon
  • Steps:
    1. Area of the hexagon base:
       • Hexagon is 6 triangles: area of one triangle using ( \frac{1}{2} \times \text{base} \times \text{height} ) = 20 yards².
       • Total for one hexagon = 120 yards², for both = 240 yards².
    2. Area of the rectangular sides:
       • One face = 800 yards², for all six faces = 4800 yards².
    3. Total surface area: 5040 yards².

Conclusion

• Summary of methods for calculating surface area for various prisms and cylinders.
• Encouragement to continue learning through similar problems.