Question

a triangular pyramid is shown below. which diagram is a net for this pyramid? what is the surface area of the triangular pyramid? square feet

Explanation:

Step1: Recall surface - area formula for triangular pyramid

The surface area \(SA\) of a triangular pyramid is the sum of the areas of its four triangular faces. The base is an equilateral triangle with side length \(a = 16\) ft and the lateral faces are isosceles triangles with base \(a = 16\) ft and height \(h_{l}=14\) ft. The area of an equilateral triangle with side length \(a\) is \(A_{base}=\frac{\sqrt{3}}{4}a^{2}\), and the area of a triangle is \(A=\frac{1}{2}bh\).

Step2: Calculate the area of the base

For the base (equilateral triangle) with \(a = 16\) ft, \(A_{base}=\frac{\sqrt{3}}{4}\times16^{2}=\frac{\sqrt{3}}{4}\times256 = 64\sqrt{3}\approx64\times1.732 = 110.848\) square - feet.

Step3: Calculate the area of one lateral face

For a lateral face with base \(b = 16\) ft and height \(h = 14\) ft, \(A_{lateral}=\frac{1}{2}\times16\times14=112\) square - feet.

Step4: Calculate the total surface area

Since there are 3 lateral faces and 1 base face, \(SA=A_{base}+3A_{lateral}\). Substitute the values: \(SA = 110.848+3\times112=110.848 + 336=446.848\approx446.85\) square - feet.

Answer:

446.85 square feet