Cuboids - Examples, Exercises and Solutions

To determine the volume of a cuboid, we apply the formula:

• Step 1: Identify the dimensions of the cuboid:
  • Length ($l$) = 12 cm
  • Width ($w$) = 8 cm
  • Height ($h$) = 5 cm
• Step 2: Apply the volume formula for a cuboid:

The formula to find the volume ($V$) of a cuboid is:

$V = l \times w \times h$

Step 3: Substitute the given dimensions into the formula and calculate: $V = 12 \times 8 \times 5$

Step 4: Perform the multiplication in stages for clarity:

First, calculate $12 \times 8 = 96$

Then multiply the result by 5: $96 \times 5 = 480$

Therefore, the volume of the cuboid is $\mathbf{480 \, \text{cm}^3}$.

To solve this problem, we'll utilize the formula for the surface area of a cuboid. The steps are as follows:

• Step 1: Identify the dimensions from the problem. The dimensions provided are $a = 3$, $b = 5$, and $c = 2$.
• Step 2: Apply the surface area formula for a cuboid. The formula is: $2(ab + bc + ac)$ where $a$, $b$, and $c$ are the dimensions of the cuboid.
• Step 3: Substitute the known values into the formula: $2(3 \cdot 5 + 5 \cdot 2 + 3 \cdot 2)$
• Step 4: Calculate each term inside the parentheses: - $a \cdot b = 3 \cdot 5 = 15$ - $b \cdot c = 5 \cdot 2 = 10$ - $a \cdot c = 3 \cdot 2 = 6$
• Step 5: Sum the results from Step 4: $15 + 10 + 6 = 31$
• Step 6: Multiply the sum by 2 to find the total surface area: $2 \times 31 = 62$

Thus, after performing the necessary calculations, the surface area of the orthohedron is $62$ square units.

To solve this problem, we'll follow these steps:

• Identify the given dimensions of the rectangular prism.
• Use the formula for volume: $V = l \times w \times h$.
• Calculate the volume by plugging in the given values.

Now, let's work through each step:
Step 1: The problem provides the dimensions of the prism: length = 3, width = 8, height = 2.
Step 2: Applying the formula, we have $V = l \times w \times h = 3 \times 8 \times 2$.
Step 3: Performing the multiplication, we obtain $V = 3 \times 8 \times 2 = 24 \times 2 = 48$.

Therefore, the volume of the rectangular prism is $48$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given dimensions of the prism.
• Step 2: Apply the formula for the volume of a rectangular prism.
• Step 3: Perform the necessary calculations.

Now, let's work through each step:
Step 1: The given dimensions are height $h = 5$, width $w = 4$, and depth $d = 9$.
Step 2: We use the formula for volume $V = h \times w \times d$.
Step 3: Plugging in our values, we have
$V = 5 \times 4 \times 9 = 180$

Therefore, the volume of the rectangular prism is $180$.

To solve this problem, we need to find the volume of the rectangular prism by following these steps:

• Step 1: Identify the given dimensions.
• Step 2: Apply the formula for the volume of a rectangular prism.
• Step 3: Plug in the values and calculate the volume.

Let's proceed with each step:

Step 1: We are given the length = 5 units, width = 8 units, and height = 12 units of the prism.

Step 2: Use the formula for the volume of a rectangular prism:
$\text{Volume} = \text{length} \times \text{width} \times \text{height}$

Step 3: Substitute the given dimensions into the formula:
$\text{Volume} = 5 \times 8 \times 12$

Now, perform the calculation:
$5 \times 8 = 40$
$40 \times 12 = 480$

Thus, the volume of the rectangular prism is $480$ cubic units.

Therefore, the correct choice from the given options, based on this calculation, is Choice 3: $480$.

Let's see what rectangles we have:

8*5

8*12

5*12

Let's review the formula for the surface area of a rectangular prism:

(length X width + length X height + width X height) * 2

Now let's substitute all this into the exercise:

(8*5+12*8+12*5)*2=
(40+96+60)*2=
196*2= 392

This is the solution!
 

Remember that the formula for the surface area of a cuboid is:

(length X width + length X height + width X height) 2

We input the known data into the formula:

2*(3*2+2*5+3*5)

2*(6+10+15)

2*31 = 62

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information
• Step 2: Apply the appropriate formula for volume
• Step 3: Perform the necessary calculations

Now, let's work through each step:

Step 1: The problem gives us the dimensions of a cuboid: length $L = 8 \, \text{cm}$, width $W = 2 \, \text{cm}$, and height $H = 4 \, \text{cm}$.

Step 2: We'll use the formula to calculate the volume of a cuboid: $V = L \times W \times H$.

Step 3: Substitute the given dimensions into the formula: $V = 8 \times 2 \times 4$ Calculate the result: $V = 16 \times 4 = 64$ Thus, the volume of the cuboid is $64 \, \text{cm}^3$.

Therefore, the solution to the problem is $64 \, \text{cm}^3$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given dimensions: length = 9 cm, width = 4 cm, height = 5 cm.
• Step 2: Apply the formula for the volume of a cuboid, $V = \text{length} \times \text{width} \times \text{height}$.
• Step 3: Calculate the value by substituting the given dimensions into the formula.

Now, let's work through each step:

Step 1: Given dimensions are:
- Length = 9 cm
- Width = 4 cm
- Height = 5 cm

Step 2: Use the formula for the volume of a cuboid:
$V = \text{length} \times \text{width} \times \text{height}$

Step 3: Substitute the values into the formula:
$V = 9 \, \text{cm} \times 4 \, \text{cm} \times 5 \, \text{cm}$

Calculate the product:
$V = 180 \, \text{cm}^3$

Therefore, the volume of the cuboid is $180 \, \text{cm}^3$.

First, we recall the formula for the surface area of a cuboid:

(width*length + height*width + height*length) *2

As in the cuboid the opposite faces are equal to each other, the given data is sufficient to arrive at a solution.

We replace the data in the formula:

(8*5+3*5+8*3) *2 =

(40+15+24) *2 =

79*2 = 

158

We identified that the faces are

3*3, 3*11, 11*3
As the opposite faces of an cuboid are equal, we know that for each face we find there is another face, therefore:

3*3, 3*11, 11*3

or

(3*3, 3*11, 11*3 ) *2

To find the surface area, we will have to add up all these areas, therefore:

(3*3+3*11+11*3 )*2

And this is actually the formula for the surface area!

We calculate:

(9+33+33)*2

(75)*2

150

Every cuboid is made up of rectangles. These rectangles are the faces of the cuboid.

As we know that in a rectangle the parallel faces are equal to each other, we can conclude that for each face found there will be two rectangles.

Let's first look at the face painted orange,

It has width and height, 5 and 3, so we already know that they are two rectangles of size 5x6

Now let's look at the side faces, they also have a height of 3, but their width is 6,

And then we understand that there are two more rectangles of 3x6

Now let's look at the top and bottom faces, we see that their dimensions are 5 and 6,

Therefore, there are two more rectangles that are size 5x6

That is, there are
2 rectangles 5X6

2 rectangles 3X5

2 rectangles 6X3

Let's go through the options:

A - In this option, we can observe that there are two flaps on the same side.

If we try to turn this net into a box, we should obtain a box where on one side there are two faces one on top of the other while the other side is "open",
meaning this net cannot be turned into a complete and full box.

B - This net looks valid at first glance, but we need to verify that it matches the box we want to draw.

In the original box, we see that we have four flaps of size 9*4, and only two flaps of size 4*4,
if we look at the net we can see that the situation is reversed, there are four flaps of size 4*4 and two flaps of size 9*4,
therefore we can conclude that this net is not suitable.

C - This net at first glance looks valid, it has flaps on both sides so it will close into a box.

Additionally, it matches our drawing - it has four flaps of size 9*4 and two flaps of size 4*4.

Therefore, we can conclude that this net is indeed the correct net.

D - In this net we can see that there are two flaps on the same side, therefore this net will not succeed in becoming a box if we try to create it.

The formula to calculate the volume of a cuboid is:

height*length*width

We replace the data in the formula:  

3*5*7

7*5 = 35

35*3 = 105

To solve this problem, we'll calculate the volume of the cuboid using the given dimensions:

• Step 1: Identify the dimensions
• Step 2: Apply the volume formula for a cuboid
• Step 3: Calculate the volume

Let's work through these steps:

Step 1: From the diagram, we are informed of two dimensions directly: the width $w = 5$ and the height $h = 4$. The diagram also indicates the horizontal length (along the base) is $l = 9$.

Step 2: To find the volume of the cuboid, we use the formula:
$\text{Volume} = \text{length} \times \text{width} \times \text{height}.$

Step 3: Substituting the identified dimensions into the formula, we have:
$\text{Volume} = 9 \times 5 \times 4.$

Calculating this, we find:
$9 \times 5 = 45,$
$45 \times 4 = 180.$

Therefore, the volume of the cuboid is $180$ cubic units.

This corresponds to choice \#4: 180.