Total Surface Area of Solids

Overview

• Topic: Total Surface Area of 3D Shapes.
• Method: Find area of each face, then add all face areas.
• Shapes covered: Cuboid, prism with triangular faces, prism with trapezium faces.
• Units: Areas given in cm² or m² as specified per example.

Cuboid Example 1 (15 × 10 × 5 cm)

• Faces: 6 rectangles (front/back, top/bottom, two sides).
• Areas:
  • Front/back: 15 × 5 = 75 cm² (each)
  • Top/bottom: 15 × 10 = 150 cm² (each)
  • Sides: 10 × 5 = 50 cm² (each)
• Total surface area calculation:
  • 75 + 75 + 150 + 150 + 50 + 50 = 550 cm²

Cuboid Practice (6 × 4 × 2 m)

• Faces:
  • Front/back: 6 × 2 = 12 m² (each)
  • Top/bottom: 6 × 4 = 24 m² (each)
  • Sides: 4 × 2 = 8 m² (each)
• Total surface area:
  • 12 + 12 + 24 + 24 + 8 + 8 = 88 m²

Triangular Prism Example (triangle base 5, height 12; rectangular faces 5×10, 10×13, 12×10) in cm

• Faces:
  • Front/back: triangle area = 1/2 × base × height = 1/2 × 5 × 12 = 30 cm² (each)
  • Bottom: 5 × 10 = 50 cm²
  • Slanted side: 10 × 13 = 130 cm²
  • Straight side: 12 × 10 = 120 cm²
• Total surface area:
  • 30 + 30 + 50 + 130 + 120 = 360 cm²

Triangular Prism Practice (triangle base 4, height 3; rectangular faces 3×20, 20×5, 4×20) in m

• Faces:
  • Front/back: 1/2 × 4 × 3 = 6 m² (each)
  • Bottom: 3 × 20 = 60 m²
  • Slanted side: 20 × 5 = 100 m²
  • Straight side: 4 × 20 = 80 m²
• Total surface area:
  • 6 + 6 + 60 + 100 + 80 = 252 m²

Trapezium-Front Prism Example 1 (parallel sides 6, 9; height 4; other faces: sides 5×8, top 6×8, bottom 9×8) in cm

• Front/back: trapezium area = 1/2 × (6 + 9) × 4 = 30 cm² (each)
• Sides: 5 × 8 = 40 cm² (each)
• Top: 6 × 8 = 48 cm²
• Bottom: 9 × 8 = 72 cm²
• Total surface area:
  • 30 + 30 + 40 + 40 + 48 + 72 = 260 cm²

Trapezium-Front Prism Practice (parallel sides 7, 11; height 7; other faces: top 7×5, bottom 11×5, slanted 8×5, straight 7×5) in cm

• Front/back: 1/2 × (7 + 11) × 7 = 63 cm² (each)
• Top: 7 × 5 = 35 cm²
• Bottom: 11 × 5 = 55 cm²
• Slanted side: 8 × 5 = 40 cm²
• Straight side: 7 × 5 = 35 cm²
• Total surface area:
  • 63 + 63 + 35 + 55 + 40 + 35 = 291 cm²

Final Practice Question 1 (cuboid 8 × 5 × 4 m)

• Faces:
  • Front/back: 8 × 4 = 32 m² (each)
  • Top/bottom: 8 × 5 = 40 m² (each)
  • Sides: 4 × 5 = 20 m² (each)
• Total surface area:
  • 32 + 32 + 40 + 40 + 20 + 20 = 184 m²

Final Practice Question 2 (trapezium front with parallel sides 6, 10; distance 5; other faces: sides 6×9, top 6×9, bottom 10×9) in cm

• Front/back: 1/2 × (6 + 10) × 5 = 40 cm² (each)
• Sides: 6 × 9 = 54 cm² (each)
• Top: 6 × 9 = 54 cm²
• Bottom: 10 × 9 = 90 cm²
• Total surface area:
  • 40 + 40 + 54 + 54 + 54 + 90 = 332 cm²

Key Terms and Formulas

• Total Surface Area: sum of areas of all faces.
• Rectangle area: length × width.
• Triangle area: 1/2 × base × height.
• Trapezium area: 1/2 × (sum of parallel sides) × distance between them.

Action Items / Next Steps

• Practice identifying all faces before calculating areas.
• Label units (cm² or m²) for each area and final total.
• Check symmetry: multiply identical face areas by 2 to avoid repetition.
• Apply formulas accurately for triangles and trapezia when those faces appear.