Question

find the surface area of the polyhedra below. find the surface area of a cube with an edge length of 5 cm?

Explanation:

Step1: Calculate the surface area of the cube

The formula for the surface area of a cube is \(S = 6a^{2}\), where \(a\) is the edge - length. Given \(a = 5\mathrm{cm}\).
Substitute \(a = 5\) into the formula: \(S=6\times5^{2}\).
First, calculate \(5^{2}=25\). Then \(6\times25 = 150\mathrm{cm}^{2}\).

Step2: Calculate the surface area of the polyhedron (using the formula for the surface area of a triangular prism \(S=2B + Ph\))

1. Find the area of the base \(B\):

The base is a right - triangle with \(b = 4\mathrm{in}\) and \(h = 3\mathrm{in}\). Using the formula \(B=\frac{1}{2}bh\), we have \(B=\frac{1}{2}\times4\times3=6\mathrm{in}^{2}\).

1. Find the perimeter of the base \(P\):

The sides of the base triangle are \(3\mathrm{in}\), \(4\mathrm{in}\), and \(5\mathrm{in}\). So \(P=3 + 4+5=12\mathrm{in}\).

1. Find the height of the prism \(h\) (the length of the prism):

\(h = 8\mathrm{in}\).

1. Calculate the surface area \(S\):

Using the formula \(S = 2B+Ph\), substitute \(B = 6\), \(P = 12\), and \(h = 8\).
\(S=2\times6+12\times8\).
First, \(2\times6 = 12\) and \(12\times8=96\). Then \(S=12 + 96=108\mathrm{in}^{2}\).

Answer:

The surface area of the cube is \(150\mathrm{cm}^{2}\) and the surface area of the polyhedron (triangular prism) is \(108\mathrm{in}^{2}\).