Question

a prism has a height of 4 centimeters. its base is shown below. what is the surface area of the prism? square centimeters

Explanation:

Step1: Calculate the area of the triangular base

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\). Here, the base of the triangle \(b = 5\) cm and the height \(h=12\) cm.

$$A_{base}=\frac{1}{2}\times5\times12= 30\space cm^{2}$$

Step2: Calculate the perimeter of the triangular base

The perimeter \(P\) of the triangle is the sum of its side - lengths. The side - lengths are \(a = 5\) cm, \(b = 12\) cm, and \(c = 13\) cm.

$$P=5 + 12+13=30\space cm$$

Step3: Calculate the lateral surface area

The formula for the lateral surface area of a prism is \(A_{lateral}=P\times h\), where \(P\) is the perimeter of the base and \(h\) is the height of the prism. Here, \(P = 30\) cm and \(h = 4\) cm.

$$A_{lateral}=30\times4=120\space cm^{2}$$

Step4: Calculate the total surface area

The formula for the total surface area of a prism is \(A_{total}=A_{lateral}+2\times A_{base}\)

$$A_{total}=120 + 2\times30$$

$$A_{total}=120+60$$

$$A_{total}=180\space cm^{2}$$

Answer:

180