Volume and Surface Area of a Rectangular Solid
For a rectangular solid with length
L, width
W, and height
H:
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
14 cm, height
17 cm, and width
9 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. |
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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 V = volume |
Step 4. Translate.
Write the appropriate formula.
Substitute. |
V=LWH
V=14cm⋅9cm⋅17cm |
Step 5. Solve the equation. |
V=2,142cm3 |
Step 6. Check
We leave it to you to check your calculations. |
|
Step 7. Answer the question. |
The volume is 2,142 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 S = surface area |
Step 4. Translate.
Write the appropriate formula.
Substitute. |
S=2LH+2LW+2WH
S=2(14cm⋅17cm)+2(14cm⋅9cm)+2(9cm⋅17cm) |
Step 5. Solve the equation. |
S=1,034cm2 |
Step 6. Check: Double-check with a calculator. |
|
Step 7. Answer the question. |
The surface area is 1,034 square centimeters. |
example
A rectangular crate has a length of
30 inches, width of
25 inches, and height of
20 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. |
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1. |
|
Step 2. Identify what you are looking for. |
The volume of the crate |
Step 3. Name. Choose a variable to represent it. |
Let V = volume |
Step 4. Translate.
Write the appropriate formula.
Substitute. |
V=LWH
V=30in⋅25in⋅20in |
Step 5. Solve the equation. |
V=15,000in3 |
Step 6. Check: Double check your math. |
|
Step 7. Answer the question. |
The volume is 15,000 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 S = surface area |
Step 4. Translate.
Write the appropriate formula.
Substitute. |
S=2LH+2LW+2WH
S=2(30in⋅20in)+2(30in⋅25in)+2(25in⋅20in) |
Step 5. Solve the equation. |
S=3,700in2 |
Step 6. Check: Check it yourself! |
|
Step 7. Answer the question. |
The surface area is 3,700 square inches. |
Volume and Surface Area of a Cube
For any cube with sides of length
s,
example
A cube is
2.5 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. |
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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. |
V=s3 |
Step 5. Solve. Substitute and solve. |
V=(2.5in)3
V=15.625in3 |
Step 6. Check: Check your work. |
|
Step 7. Answer the question. |
The volume is 15.625 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. |
S=6s2 |
Step 5. Solve. Substitute and solve. |
S=6⋅(2.5in)2
S=37.5in2 |
Step 6. Check: The check is left to you. |
|
Step 7. Answer the question. |
The surface area is 37.5 square inches. |
example
A notepad cube measures
2 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. |
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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. |
V=s3 |
Step 5. Solve the equation. |
V=(2in)3
V=8in3 |
Step 6. Check: Check that you did the calculations
correctly. |
|
Step 7. Answer the question. |
The volume is 8 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. |
S=6s2 |
Step 5. Solve the equation. |
S=6⋅(2in)2
S=24in2 |
Step 6. Check: The check is left to you. |
|
Step 7. Answer the question. |
The surface area is 24 square inches. |
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