Calculate Volume of Connected Rectangular Prisms: 5×9×2 Composite Shape

Q: How do I know which dimensions go with which prism?

Look carefully at the diagram! The square base prism has equal sides (5×5) and height 9. The attached smaller prism has different dimensions: length 3, width 2, and height 4.

Q: Why don't I subtract one volume from the other?

These are separate prisms attached together, not one shape inside another. When shapes are connected like this, you add their volumes to find the total space they occupy.

Q: What's the difference between a square base and regular rectangular prism?

A square base prism has two equal sides forming the base (like 5×5), so you calculate $ side^2 \times height $. A regular rectangular prism uses $ length \times width \times height $ with three different measurements.

Q: How can I be sure I read the measurements correctly?

Follow the colored labels in the diagram! Each measurement is marked in a different color and points to the specific edge it measures. Double-check by counting the dimensions: you should have exactly 5 measurements total.

Q: What if I get a different answer when I add the volumes?

Go back and recalculate each volume separately. Common errors include using wrong dimensions (like 5×9×2 instead of $ 5^2 \times 9 $) or forgetting to square the base of the square prism.

Volume Addition with Composite Shapes

A rectangular prism with a square base is attached to a rectangular prism as shown below.

Calculate the volume of the new shape using the data provided.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the volume of the shared body

00:04 We'll use the formula for calculating box volume

00:07 Height times length times width

00:10 The box base is a square according to the data, so sides are equal

00:14 We'll substitute appropriate values and solve for the volume

00:17 Let's start with box 1

00:21 This is the volume of box 1

00:24 Now let's calculate the volume of box 2 in the same way

00:36 This is the volume of box 2

00:41 We'll sum the volumes to find the shared body volume

00:46 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A rectangular prism with a square base is attached to a rectangular prism as shown below.

Calculate the volume of the new shape using the data provided.

Step-by-step solution

To solve this problem, we will calculate the volume of each rectangular prism separately and then sum these volumes:

Rectangular Prism 1: The prism with a square base has dimensions as follows:

Side length of the square base = 5 units
Height = 9 units

The volume of a rectangular prism with a square base is given by $V = \text{side}^2 \times \text{height}$ .
Substituting the given values: $V_1 = 5^2 \times 9 = 25 \times 9 = 225 \text{ cubic units}$ .
Rectangular Prism 2: The second prism's dimensions:

Length = 3 units
Width = 2 units
Height = 4 units

The volume of this rectangular prism is calculated as $V = \text{length} \times \text{width} \times \text{height}$ .
Substituting the given values: $V_2 = 3 \times 2 \times 4 = 24 \text{ cubic units}$ .

Finally, adding the volumes of the two prisms gives us the total volume:

$V_{\text{total}} = V_1 + V_2 = 225 + 24 = 249 \text{ cubic units}$ .

Therefore, the volume of the new shape is $249$ cubic units.

Final Answer

$249$

Key Points to Remember

Essential concepts to master this topic

Rule: Calculate each prism separately then add volumes together
Technique: Square base prism: $5^2 \times 9 = 225$ , regular prism: $3 \times 2 \times 4 = 24$
Check: Total volume should equal sum: $225 + 24 = 249$ cubic units ✓

Common Mistakes

Avoid these frequent errors

Mixing up dimensions between the two prisms
Don't use the 5×9×2 measurements for both shapes = wrong calculation! The square base prism has sides 5 and height 9, while the attached prism has dimensions 3×2×4. Always identify which measurements belong to which prism first.

Practice Quiz

Test your knowledge with interactive questions

A rectangular prism has a base measuring 5 units by 8 units.

The height of the prism is 12 units.

Calculate its volume.

FAQ

Everything you need to know about this question

How do I know which dimensions go with which prism?

Look carefully at the diagram! The square base prism has equal sides (5×5) and height 9. The attached smaller prism has different dimensions: length 3, width 2, and height 4.

Why don't I subtract one volume from the other?

These are separate prisms attached together, not one shape inside another. When shapes are connected like this, you add their volumes to find the total space they occupy.

What's the difference between a square base and regular rectangular prism?

A square base prism has two equal sides forming the base (like 5×5), so you calculate $side^2 \times height$ . A regular rectangular prism uses $length \times width \times height$ with three different measurements.

How can I be sure I read the measurements correctly?

Follow the colored labels in the diagram! Each measurement is marked in a different color and points to the specific edge it measures. Double-check by counting the dimensions: you should have exactly 5 measurements total.

What if I get a different answer when I add the volumes?

Go back and recalculate each volume separately. Common errors include using wrong dimensions (like 5×9×2 instead of $5^2 \times 9$ ) or forgetting to square the base of the square prism.

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