Question

what is the surface area of the shape? 2in 2in 8in in.²

Explanation:

Step1: Identify the shape as a triangular - based pyramid

The surface area of a triangular - based pyramid is the sum of the areas of its faces.

Step2: Area of the base triangle

The base is an equilateral triangle with side length \(s = 8\) in. The area of an equilateral triangle \(A_{base}=\frac{\sqrt{3}}{4}s^{2}=\frac{\sqrt{3}}{4}\times8^{2}=\frac{\sqrt{3}}{4}\times64 = 16\sqrt{3}\text{ in}^2\approx16\times1.732 = 27.712\text{ in}^2\).

Step3: Area of each lateral face

Each lateral face is a triangle with base \(b = 8\) in and height \(h = 2\) in. The area of a triangle is \(A=\frac{1}{2}bh\). For each lateral face, \(A_{lateral}=\frac{1}{2}\times8\times2=8\text{ in}^2\).

Step4: Calculate the total surface area

There are 3 lateral faces and 1 base face. \(A_{total}=A_{base}+3A_{lateral}\). Substituting the values, \(A_{total}=16\sqrt{3}+3\times8=16\sqrt{3}+24\approx27.712 + 24=51.712\text{ in}^2\).

Answer:

\(51.712\) (rounded to three decimal places)