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a regular prism with a pentagonal base has a height of 12 cm12\,\text{cm}. the apothem of the base measures 2.75 cm2.75\,\text{cm} and its sides, 4 cm4\,\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 cm12\,\text{cm}. the apothem of the base measures 2.75 cm2.75\,\text{cm} and its sides, 4 cm4\,\text{cm}. What is the total area of ​​the prism?
  1. 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 PP of the base times the height hh of the prism. The perimeter of a pentagonal base is the sum of the lengths of its five sides.\newlineCalculation: P=5×side_length=5×4 cm=20 cmP = 5 \times \text{side\_length} = 5 \times 4\,\text{cm} = 20\,\text{cm}
  2. Calculate Lateral Area: Now we calculate the lateral area using the perimeter and the height of the prism.\newlineCalculation: LA=P×h=20 cm×12 cm=240 cm2LA = P \times h = 20 \, \text{cm} \times 12 \, \text{cm} = 240 \, \text{cm}^2
  3. Calculate Base Area: Next, we need to calculate the area of one pentagonal base. The area AA of a regular pentagon can be calculated using the formula A=12⋅P⋅aA = \frac{1}{2} \cdot P \cdot a, where PP is the perimeter and aa is the apothem.\newlineCalculation: A=12⋅P⋅a=12⋅20 cm⋅2.75 cm=27.5 cm2A = \frac{1}{2} \cdot P \cdot a = \frac{1}{2} \cdot 20 \, \text{cm} \cdot 2.75 \, \text{cm} = 27.5 \, \text{cm}^2
  4. Calculate Total Base Area: Since the prism has two identical pentagonal bases, we need to multiply the area of one base by 22 to get the total area of both bases.\newlineCalculation: Total area of both bases = A×2=27.5 cm2×2=55 cm2A \times 2 = 27.5 \, \text{cm}^2 \times 2 = 55 \, \text{cm}^2
  5. 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.\newlineCalculation: Total surface area = LA+Total area of both bases=240 cm2+55 cm2=295 cm2LA + \text{Total area of both bases} = 240 \, \text{cm}^2 + 55 \, \text{cm}^2 = 295 \, \text{cm}^2

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