Question

Draw a rectangular prism that is 4 centimeters by 5 centimeters by 8 centimeters. Find the surface area of the prism.

Short answer

The surface area is 184 cm².

Step-by-step solution

Understand the dimensions of the rectangular prism

The rectangular prism has three dimensions: the length is 8 cm, the width is 5 cm, and the height is 4 cm. We need to use these measurements to calculate the surface area.

Determine the formula for the surface area of a rectangular prism

The surface area of a rectangular prism is calculated using the formula: \[ SA = 2(lw + lh + wh) \]where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

Substitute the given dimensions into the formula

Using the given dimensions, where \( l = 8 \) cm, \( w = 5 \) cm, and \( h = 4 \) cm, substitute them into the surface area formula:\[ SA = 2((8\times5) + (8\times4) + (5\times4)) \]

Calculate the individual areas of each pair of faces

Calculate the area for each pair of faces:- The area of the base faces (length and width): \[ 8 \times 5 = 40 \, \text{cm}^2 \]- The area of the side faces (length and height): \[ 8 \times 4 = 32 \, \text{cm}^2 \]- The area of the front and back faces (width and height): \[ 5 \times 4 = 20 \, \text{cm}^2 \]

Sum the calculated areas and complete the surface area formula

Now, add the three calculated areas and multiply by 2 to get the total surface area:\[ SA = 2(40 + 32 + 20) \]Calculate the sum inside the parentheses:\[ 40 + 32 + 20 = 92 \]Then, multiply by 2:\[ SA = 2 \times 92 = 184 \, \text{cm}^2 \]

Conclusion

The total surface area of the rectangular prism is 184 square centimeters.

Key concepts

Rectangular Prism Dimensions

A rectangular prism is a three-dimensional shape, much like a box, with six faces that are rectangles. It's important to understand its dimensions, as they are foundational to geometry problem solving and surface area calculations. In the case of the exercise, the rectangular prism has:

• Length (\( l \)): 8 centimeters
• Width (\( w \)): 5 centimeters
• Height (\( h \)): 4 centimeters

These dimensions determine not only the shape's structure but also how large each rectangular face is. Remember, the length, width, and height may not always correspond to the orientation of the shape as seen, thus understanding and identifying each dimension correctly is crucial.

Geometry Problem Solving

Solving geometry problems often involves breaking down a problem into steps or parts that can be more easily managed. When tackling rectangular prism problems, consider:

• Identifying all given measurements, which helps formulate how each dimension interacts to form the shape.
• Visualizing the 3D form in your mind or drawing it out, which aids in understanding and ensuring all surfaces are accounted for during calculations.
• Applying the correct mathematical formulas, a common challenge in geometry, because each shape has unique properties that define how calculations should be made.

In our exercise, the surface area formula must be correctly applied based on the accurate interpretation of the prism’s dimensions. This step-by-step approach helps ensure precision and accuracy in solutions.

Surface Area Calculation

The surface area calculation of a rectangular prism involves finding the total area of all six rectangular faces. The formula to calculate this is:\[ SA = 2(lw + lh + wh) \]Where \( l \) is the length, \( w \) is the width, and \( h \) is the height.Here's how to proceed with the calculation:

• First, calculate the area of each distinct pair of opposite faces, as each pair will have identical measurements:
• Base faces (length × width): \( 8 \times 5 = 40 \text{ cm}^2 \)
• Side faces (length × height): \( 8 \times 4 = 32 \text{ cm}^2 \)
• Front and Back faces (width × height): \( 5 \times 4 = 20 \text{ cm}^2 \)
• Next, add these areas together: \( 40 + 32 + 20 = 92 \text{ cm}^2 \)
• Finally, multiply by 2, because each face has an identical opposite face: \( SA = 2 \times 92 = 184 \text{ cm}^2 \)

Thus, the total surface area of the rectangular prism is 184 square centimeters. This systematic approach ensures that no face is left out in the calculation and the correct area is computed efficiently.