Question

a rectangular prism has a length of 8 cm, a width of 5 cm, and a height of 3 cm.

1. find the area of the rectangular face that measures 8 cm by 5 cm.
2. use your answer to help calculate the prisms volume. explain how the face area relates to finding the volume.
3. finally, calculate the surface area of the entire prism. show all of your steps and explain your reasoning for how you determined which faces to include.

extension (dok 4):
if the prism is painted on the outside and then cut into 1 cm³ cubes, how many of the small cubes will have paint on at least one face? explain your reasoning.

Explanation:

Step1: Calculate face - area

The area formula for a rectangle is $A = l\times w$. For a rectangle with $l = 8$ cm and $w = 5$ cm, we have $A=8\times5 = 40$ $cm^{2}$.

Step2: Calculate volume and explain

The volume formula of a rectangular prism is $V=A_{base}\times h$. Here, the base can be the face with area $A = 40$ $cm^{2}$ and $h = 3$ cm. So $V=40\times3=120$ $cm^{3}$. The face - area is the base - area, and multiplying it by the height gives the volume as the volume of a prism is the amount of space it occupies, and we are stacking up layers with area equal to the base - area to a height $h$.

Step3: Calculate surface - area

A rectangular prism has 6 faces. The pairs of opposite faces have the same area.
The three pairs of faces have areas:

• $A_1 = 8\times5=40$ $cm^{2}$ (two such faces, total area $2\times40 = 80$ $cm^{2}$)
• $A_2=8\times3 = 24$ $cm^{2}$ (two such faces, total area $2\times24=48$ $cm^{2}$)
• $A_3 = 5\times3=15$ $cm^{2}$ (two such faces, total area $2\times15 = 30$ $cm^{2}$)

The total surface area $S=80 + 48+30=158$ $cm^{2}$. We include all 6 faces because the surface - area is the total area of the outer surface of the prism.

Step4: Calculate number of painted cubes

The total number of $1$ $cm^{3}$ cubes in the prism is $n = 8\times5\times3=120$.
The number of non - painted cubes forms a smaller rectangular prism inside. The dimensions of the non - painted prism are $(8 - 2)\times(5 - 2)\times(3 - 2)=6\times3\times1 = 18$.
The number of cubes with paint on at least one face is $120-18 = 102$. We subtract the non - painted cubes from the total number of cubes because non - painted cubes are those that are completely inside the prism and not exposed to the paint.

Answer:

1. $40$ $cm^{2}$
2. Volume is $120$ $cm^{3}$. The face - area is the base - area and multiplying by height gives volume.
3. $158$ $cm^{2}$

Extension: $102$