Volume and Surface Area of a Rectangular Solid
For a rectangular solid with length [latex]L[/latex], width [latex]W[/latex], and height [latex]H:[/latex]
Doing the Manipulative Mathematics activity "Painted Cube" will help you develop a better understanding of volume and surface area.
example
For a rectangular solid with length [latex]14[/latex] cm, height [latex]17[/latex] cm, and width [latex]9[/latex] cm, find 1. the volume and 2. the surface area.
Solution
Step 1 is the same for both 1. and 2., so we will show it just once.
Step 1. Read the problem. Draw the figure and
label it with the given information. 

1. 

Step 2. Identify what you are looking for. 
The volume of the rectangular solid 
Step 3. Name. Choose a variable to represent it. 
Let [latex]V[/latex] = volume 
Step 4. Translate.
Write the appropriate formula.
Substitute. 
[latex]V=LWH[/latex]
[latex]V=\mathrm{14}cm\cdot 9cm\cdot 17cm[/latex] 
Step 5. Solve the equation. 
[latex]V=2,142cm^3[/latex] 
Step 6. Check
We leave it to you to check your calculations. 

Step 7. Answer the question. 
The volume is [latex]\text{2,142}[/latex] cubic centimeters. 
2. 

Step 2. Identify what you are looking for. 
the surface area of the solid 
Step 3. Name. Choose a variable to represent it. 
Let [latex]S[/latex] = surface area 
Step 4. Translate.
Write the appropriate formula.
Substitute. 
[latex]S=2LH+2LW+2WH[/latex]
[latex]S=2\left(14cm\cdot 17cm\right)+2\left(14cm\cdot 9cm\right)+2\left(9cm\cdot 17cm\right)[/latex] 
Step 5. Solve the equation. 
[latex]S=1,034cm^2[/latex] 
Step 6. Check: Doublecheck with a calculator. 

Step 7. Answer the question. 
The surface area is [latex]1,034[/latex] square centimeters. 
example
A rectangular crate has a length of [latex]30[/latex] inches, width of [latex]25[/latex] inches, and height of [latex]20[/latex] inches. Find 1. its volume and 2. its surface area.
Answer:
Solution
Step 1 is the same for both 1. and 2., so we will show it just once.
Step 1. Read the problem. Draw the figure and
label it with the given information. 

1. 

Step 2. Identify what you are looking for. 
The volume of the crate 
Step 3. Name. Choose a variable to represent it. 
Let [latex]V[/latex] = volume 
Step 4. Translate.
Write the appropriate formula.
Substitute. 
[latex]V=LWH[/latex]
[latex]V=30in\cdot 25in\cdot 20in[/latex] 
Step 5. Solve the equation. 
[latex]V=15,000in^3[/latex] 
Step 6. Check: Double check your math. 

Step 7. Answer the question. 
The volume is [latex]15,000[/latex] cubic inches. 
2. 

Step 2. Identify what you are looking for. 
The surface area of the crate 
Step 3. Name. Choose a variable to represent it. 
Let [latex]S[/latex] = surface area 
Step 4. Translate.
Write the appropriate formula.
Substitute. 
[latex]S=2LH+2LW+2WH[/latex]
[latex]S=2\left(30in\cdot 20in\right)+2\left(30in\cdot 25in\right)+2\left(25in\cdot 20in\right)[/latex] 
Step 5. Solve the equation. 
[latex]S=3,700in^2[/latex] 
Step 6. Check: Check it yourself! 

Step 7. Answer the question. 
The surface area is [latex]3,700[/latex] square inches. 
Volume and Surface Area of a Cube
For any cube with sides of length [latex]s[/latex],
example
A cube is [latex]2.5[/latex] inches on each side. Find 1. its volume and 2. its surface area.
Solution
Step 1 is the same for both 1. and 2., so we will show it just once.
Step 1. Read the problem. Draw the figure and
label it with the given information. 

1. 

Step 2. Identify what you are looking for. 
The volume of the cube 
Step 3. Name. Choose a variable to represent it. 
Let V = volume 
Step 4. Translate.
Write the appropriate formula. 
[latex]V={s}^{3}[/latex] 
Step 5. Solve. Substitute and solve. 
[latex]V={\left(2.5in\right)}^{3}[/latex]
[latex]V=15.625in^3[/latex] 
Step 6. Check: Check your work. 

Step 7. Answer the question. 
The volume is [latex]15.625[/latex] cubic inches. 
2. 

Step 2. Identify what you are looking for. 
The surface area of the cube 
Step 3. Name. Choose a variable to represent it. 
Let S = surface area 
Step 4. Translate.
Write the appropriate formula. 
[latex]S=6{s}^{2}[/latex] 
Step 5. Solve. Substitute and solve. 
[latex]S=6\cdot {\left(2.5in\right)}^{2}[/latex]
[latex]S=37.5in^2[/latex] 
Step 6. Check: The check is left to you. 

Step 7. Answer the question. 
The surface area is [latex]37.5[/latex] square inches. 
example
A notepad cube measures [latex]2[/latex] inches on each side. Find 1. its volume and 2. its surface area.
Answer:
Solution
Step 1. Read the problem. Draw the figure and
label it with the given information. 

1. 

Step 2. Identify what you are looking for. 
The volume of the cube 
Step 3. Name. Choose a variable to represent it. 
Let V = volume 
Step 4. Translate.
Write the appropriate formula. 
[latex]V={s}^{3}[/latex] 
Step 5. Solve the equation. 
[latex]V=({2in})^{3}[/latex]
[latex]V=8in^3[/latex] 
Step 6. Check: Check that you did the calculations
correctly. 

Step 7. Answer the question. 
The volume is [latex]8[/latex] cubic inches. 
2. 

Step 2. Identify what you are looking for. 
The surface area of the cube 
Step 3. Name. Choose a variable to represent it. 
Let S = surface area 
Step 4. Translate.
Write the appropriate formula. 
[latex]S=6{s}^{2}[/latex] 
Step 5. Solve the equation. 
[latex]S=6\cdot ({2in})^{2}[/latex]
[latex]S=24 in^2[/latex] 
Step 6. Check: The check is left to you. 

Step 7. Answer the question. 
The surface area is [latex]24[/latex] square inches. 
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