What is the total surface area of the rectangular prism shown below?

Question image

Understand the Problem

The question is asking for the calculation of the total surface area of a rectangular prism with given dimensions. To solve it, we will use the formula for the surface area of a rectangular prism: SA = 2(lw + lh + wh), where l is the length, w is the width, and h is the height.

Answer

The total surface area of the rectangular prism is $238 \, \text{cm}^2$.
Answer for screen readers

The total surface area of the rectangular prism is $238 , \text{cm}^2$.

Steps to Solve

  1. Identify the dimensions of the prism

From the problem, we know the dimensions of the rectangular prism:

  • Length ($l$) = 13 cm
  • Width ($w$) = 3 cm
  • Height ($h$) = 5 cm
  1. Use the surface area formula

The formula to calculate the surface area (SA) of a rectangular prism is:

$$ SA = 2(lw + lh + wh) $$

We will calculate each part of the formula separately.

  1. Calculate the area of each pair of sides

First, calculate the area of each pair of sides:

  • Length × Width: $$ lw = 13 , \text{cm} \times 3 , \text{cm} = 39 , \text{cm}^2 $$

  • Length × Height: $$ lh = 13 , \text{cm} \times 5 , \text{cm} = 65 , \text{cm}^2 $$

  • Width × Height: $$ wh = 3 , \text{cm} \times 5 , \text{cm} = 15 , \text{cm}^2 $$

  1. Sum the areas

Now, add the results from the previous calculations:

$$ lw + lh + wh = 39 + 65 + 15 = 119 , \text{cm}^2 $$

  1. Calculate the total surface area

Finally, multiply the sum by 2 to get the total surface area:

$$ SA = 2(119 , \text{cm}^2) = 238 , \text{cm}^2 $$

The total surface area of the rectangular prism is $238 , \text{cm}^2$.

More Information

The total surface area represents the total area that the surface of the prism occupies. It can be useful for various practical applications, such as determining the amount of material needed to cover or paint the prism.

Tips

  • Forgetting to multiply the calculated area by 2 for the final surface area.
  • Confusing dimensions when substituting into the formula, which can lead to incorrect calculations.

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