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Calculating Surface Area of Triangular Prisms

Apr 23, 2025

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Surface Area of a Triangular Prism

Key Concepts

  • Triangular Prism: A three-dimensional geometric shape with two triangular bases and three rectangular lateral faces.
  • Surface Area: The total area covering the surface of a 3D object.
  • Lateral Area: The sum of the areas of the vertical faces (not including the bases).

Problem 1: Basic Calculation

  1. Given Measurements:

    • CF = 30, AB = 10, AC = 21, EF = 17, height of triangle = 8.

  2. Lateral Area Calculation:

    • Formula: ( \text{Perimeter of the base} \times \text{Height of prism} ).
    • Perimeter of base (triangle ABC): AB + BC + AC = 48.
    • Height of prism = CF = 30.
    • Lateral area = 48 × 30 = 1440 square units.

  3. Total Surface Area Calculation:

    • Formula: ( \text{Area of the base} + \text{Lateral area} ).
    • Area of one triangle (base): ( \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 21 \times 8 = 84 ).
    • Total area of two triangles: 2 × 84 = 168.
    • Total surface area = 168 + 1440 = 1608 square units.

Problem 2: Isosceles Triangle Prism

  1. Given Measurements:

    • AB = BC = 13 (isosceles triangle), AC = 10, CF = 15, height = 12.

  2. Lateral Area Calculation:

    • Perimeter of base: 10 + 13 + 13 = 36.
    • Lateral area = 36 × 15 = 540 square inches.

  3. Total Surface Area Calculation:

    • Area of one triangle: ( \frac{1}{2} \times 10 \times 12 = 60 ).
    • Total area of two triangles: 2 × 60 = 120.
    • Total surface area = 120 + 540 = 660 square inches.

Problem 3: Triangular Prism with All Sides Given

  1. Given Measurements:

    • Triangle sides 6, 7, 8; height of prism = 15.

  2. Lateral Area Calculation:

    • Perimeter of base: 6 + 7 + 8 = 21.
    • Lateral area = 21 × 15 = 315 square units.

  3. Total Surface Area Calculation Using Heron's Formula:

    • Semi-perimeter ( s = \frac{21}{2} = 10.5 ).
    • Using Heron's formula: Area = ( \sqrt{s(s-a)(s-b)(s-c)} ), where ( a = 6, b = 7, c = 8 ).
      • ( s-a = 4.5, s-b = 3.5, s-c = 2.5 ).
    • Area of one triangle = 20.33 square units.
    • Total area of two triangles: 2 × 20.33 = 40.66.
    • Total surface area ≈ 355.7 square units.

Conclusion

  • To find the surface area of a triangular prism, calculate and sum the areas of the two triangular bases and the lateral surface area. Use specific formulas for perimeter and Heron's formula where applicable.

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