a regular prism with a pentagonal base has a height of $12\,\text{cm}$ . the apothem of the base measures $2.75\,\text{cm}$ and its sides, $4\,\text{cm}$ . What is the total area of the prism?

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Q. a regular prism with a pentagonal base has a height of

12\,\text{cm}

. the apothem of the base measures

2.75\,\text{cm}

and its sides,

4\,\text{cm}

. What is the total area of the prism?

Calculate Perimeter: To find the total area of the prism, we need to calculate the lateral area and the area of the two pentagonal bases. The lateral area (LA) of a prism is given by the perimeter $P$ of the base times the height $h$ of the prism. The perimeter of a pentagonal base is the sum of the lengths of its five sides. $\newline$ Calculation: $P = 5 \times \text{side\_length} = 5 \times 4\,\text{cm} = 20\,\text{cm}$
Calculate Lateral Area: Now we calculate the lateral area using the perimeter and the height of the prism. $\newline$ Calculation: $LA = P \times h = 20 \, \text{cm} \times 12 \, \text{cm} = 240 \, \text{cm}^2$
Calculate Base Area: Next, we need to calculate the area of one pentagonal base. The area $A$ of a regular pentagon can be calculated using the formula $A = \frac{1}{2} \cdot P \cdot a$ , where $P$ is the perimeter and $a$ is the apothem. $\newline$ Calculation: $A = \frac{1}{2} \cdot P \cdot a = \frac{1}{2} \cdot 20 \, \text{cm} \cdot 2.75 \, \text{cm} = 27.5 \, \text{cm}^2$
Calculate Total Base Area: Since the prism has two identical pentagonal bases, we need to multiply the area of one base by $2$ to get the total area of both bases. $\newline$ Calculation: Total area of both bases = $A \times 2 = 27.5 \, \text{cm}^2 \times 2 = 55 \, \text{cm}^2$
Calculate Total Surface Area: Finally, we add the lateral area and the total area of the two bases to find the total surface area of the prism. $\newline$ Calculation: Total surface area = $LA + \text{Total area of both bases} = 240 \, \text{cm}^2 + 55 \, \text{cm}^2 = 295 \, \text{cm}^2$