Math Problem Statement

What’s the answer

The approximate height of each triangle is 12.1 ft. The side length of each triangle is 14 ft. What is the surface area of the triangular pyramid?

Solution

To solve the problem, we need to calculate the surface area of a triangular pyramid (also called a tetrahedron) where all faces are equilateral triangles.

Step-by-Step Solution:

  1. Identify the Parameters:

    • Each triangle is an equilateral triangle.
    • The height of each triangle is 12.1ft12.1 \, \text{ft}.
    • The side length of each triangle is 14ft14 \, \text{ft} (as shown at the base).
  2. Formula for the Area of an Equilateral Triangle: The area AA of an equilateral triangle with side length ss is given by the formula: A=s234A = \frac{s^2 \sqrt{3}}{4} Here, s=14fts = 14 \, \text{ft}.

  3. Calculate the Area of One Triangle: Substituting the side length: A=14234=1963484.95ft2A = \frac{14^2 \sqrt{3}}{4} = \frac{196 \sqrt{3}}{4} \approx 84.95 \, \text{ft}^2

  4. Total Surface Area: A triangular pyramid has 4 equilateral triangles. Therefore, the total surface area AtotalA_{\text{total}} is: Atotal=4×84.95ft2=339.8ft2A_{\text{total}} = 4 \times 84.95 \, \text{ft}^2 = 339.8 \, \text{ft}^2

Final Answer:

The surface area of the triangular pyramid is approximately 339.8 square feet.

Would you like more details or have any questions?

Here are some related questions you can explore:

  1. How is the formula for the area of an equilateral triangle derived?
  2. What is the volume of this triangular pyramid?
  3. How do you calculate the height of an equilateral triangle if the side length is known?
  4. What if the base was not an equilateral triangle, how would the surface area change?
  5. How does the surface area of this pyramid compare to a square pyramid with the same base length?

Tip: Always remember that in an equilateral triangle, the relationship between side length and height is useful for multiple geometric applications.

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Math Problem Analysis

Mathematical Concepts

Geometry
Surface Area
Equilateral Triangle

Formulas

Area of an equilateral triangle: A = (s^2 * sqrt(3)) / 4
Surface area of a triangular pyramid: A_total = 4 * A

Theorems

Pythagorean theorem (used to calculate triangle properties)
Equilateral triangle properties

Suitable Grade Level

Grades 9-10