How to Find the Volume of a Cube

Introduction:

To realize the fundamentals of geometry is essential not only for academic purpose but also for practical applications in various fields. One of the fundamental geometric shapes is the cube.Then a question must come,

What is a cube ?

A cube is a three-dimensional geometrical shape with six equal square faces, each meeting at right angles.

To find the volume of a cube is a straightforward process, but it requires a clear understanding of the concept and the necessary formulae. In this post, I will try to delve into the details of How to Find the Volume of a Cube, will explore its properties, and discuss its applications in our todays life.

Explanation:

Understanding the Cube:

Before I delve into the concept How to Find the Volume of a Cube, let's briefly review its properties. A cube is a special case of a rectangular prism, where all six faces are squares, and all angles are right angles. Here are some key properties of a cube:

Equal Side Lengths: All edges of a cube have the same length, denoted by l

Equal Face Areas: Each face of a cube is a square, and all faces have the same area.

Right Angles: All angles formed between adjacent faces are right angles (90 degrees).

Symmetry: A cube exhibits symmetry along its axes and diagonals.

The formula for volume of a Cube:

The volume of any three-dimensional shape represents the amount of space enclosed within it. For a cube, the volume

V can be calculated using a simple formula:

V = l³

Where:

V is the volume of the cube.

l is the length of one side of the cube (also known as the edge length).

This formula arises from the fact that a cube has three dimensions (length, breadth, and height), and all three dimensions are equal (since it's a cube). Therefore, to find the volume, we simply multiply the length of one side by itself three times (l×l×l).

Step by Step process to finding the volume of Cube:

To find the volume of a cube,we will follow these simple steps:

Step 1: We will measure the length of any one side of the Cube:

Using a ruler or any measuring tool, we will determine the length of one side of the cube. We must ensure that the measurement is accurate, as it directly affects the volume calculation.

Step 2: We will cube the length:

Once We have the length of one side (l), we will cube it by multiplying it by itself twice (l³ = l x l x l). This will give you the volume of the cube.

Step 3: We will interpret the Result

The result obtained from Step 2 represents the volume of the cube in cubic units (such as cubic centimeters, cubic meters, etc.). Interpret this value in the context of your problem or application.

Example Calculation

Let's illustrate the process with an example:

Let us,Suppose we have a cube with a side length of 5 centimeters. To find its volume, we will follow these steps:

1. We will measure the Side Length:

2. Here,l=5 cm

3. We will cube the Length:

4. ∴ V = l³

5. V = 5³

=125 cubic centimeters.

6. We will Interpret the Result: The volume of the cube is 125 cubic centimeters.

Some Problems related on Volume of Cube:

Question(1): Two cubes each of volume 27m³ are joined end to end.Find the Surface Area of resulting cuboid.

Answer: Here,

Volume of each cube = 27m³

Let each side of the cube = l m

Now, A/Q

l³ = 27

=> l = ∛27

=> l = 3

∴ Length of each side of the cube (l) = 3 m

∵ Both cube are joined end to end

So,length of the cuboid (L) will be = 2 x 3

= 6 m

breadth of the cuboid (b) = 3m

and height of the cuboid (h) = 3 m

∴ Surface Area of resulting cuboid

= 2(lb + bh + hl)

= 2( 6 x 3 + 3 x 3 + 3 x 6) [putting values]

= 2 ( 18 + 9 + 18)

= 2 x 45

= 90 m²

Question(2): Each side of a cube is 4 cm, Find the volume of the cube .

Answer: Here, Each side of the cube (l) = 4cm

∴ Volume of the cube (V) = l³

= (4)³ [ putting value]

= 4 x 4 x 4

= 64 cm³

Question(3): The volume of a cube is 125cm³, find its side.

Answer: ∵ Volume of a cube = 125cm³

Let, Each side of the cube = l cm

Now according to question,

l³ = 125

=> l = ∛125

=> l = 5

∴ Each side of the cube (l) = 5cm

Alternative Method of finding the volume of Cube:

As, the basic formula of How to Find the Volume of a Cube is straightforward, there are alternative methods we can apply depending on the information available to us.

Diagonal Method:

If we're provided with the length of a cube's diagonal (the line connecting two opposite vertices), we can use it to find the volume of the cube.

The formula of finding diagonal of cube is given by,

d = √3 l

where, d = Length of the diagonal of the cube, and

l = Length of each side of the cube

Once we’ve determined the length of one side, we can use the basic volume formula to find the volume.

Surface Area Method:

If we're given the surface area of the cube (SA), we can indirectly find the volume of the cube.The surface area of a cube with side length l is given by:

S.A = 2(lb + bh + hl),[ where, l, b and h are length, breadth and height of the cube]

=2( l x l + l x l + l x l) [ ∵ In case, of cube, l = b = h ]

= 2( l² + l² + l²)

= 2 x 3l²

= 6l²

From this equation, we can solve for

l and then use the basic volume formula to find the volume.

Now we will quickly review

What is Surface Area of a Cube?

Answer: Surface Area of a Cube is the total sum of the areas of its all faces.

Some Problems related on Surface Area of a Cube:

Question(A): What is the surface area of a cube of side 3 m.

Answer: Here,

Each side of the cube(l) = 3 m

∴ Surface Area of the cube = 6l²

= 6 x 3² [ Putting value ]

= (6 x 9) m²

= 54 m²

Question(B): The surface Area of a cube is 96 square centimeter, find its side.

Answer: Here,

Surface Area = 96 cm²

Let, Each side of the cube = l cm

Now, A/Q,

6 l² = 96

=> l² = 96 / 6

=> l² = 16

=> l = ± √16

=> l = ± 4

∴ Length of each side of the cube(l) = 4 cm [neglecting - 4 ]

Question(C): The volume of a cube is 216 cm³, find its surface area.

Answer: The volume of a cube = 216 cm³

Let, its each side = l cm

Now, A/Q

l³ = 216

=> l = ∛ 216

=> l = 6

∴ Each side(l) of the cube = 6 cm

∴Surface Area of the cube = 6l²

= 6 x 6² [Putting value]

= 6 x 36

= 216 cm²

Now we will try to know,

What is Lateral Surface Area of a Cube?

Answer: Lateral Surface Area or curve surface area of a cube can be defined as, the sum of its all four surrounding faces areas excluding its top and bottom areas.

The formula of Lateral Surface Area of a cube = 2 (l +b)h [ where, l, b and h are length, breadth and height of the cube]

= 2(l + l ) l [ ∵ In case, of cube, l = b = h ]

= 2x ( 2l) x l

= 4 l²

Some Problems related on Lateral Surface Area of a Cube.

Question(a): Find the lateral surface area of a cube whose edge is 4 cm.

Answer: Here,

Edge of the cube(l) = 4 cm

∴ Lateral surface area of the cube = 4l²

= 4 x 4² [putting value]

= 4 x 16

= 64 cm³

Question(b): The curved surface area of a cube is 400 square meter,find its curved faces.

Answer: Here,

Curved surface area of the cube = 400 m²

Let, length of its each curved face = l m

Now A/Q,

4l² = 400

=> l² = 400 / 4

=> l² = 100

=> l = ± 10

∴ Length of its each curved face (l) = 10m [neglecting -10]

Question(c): What is the lateral surface area of a cube whose side is 7cm.

Answer: Here,

side of the cube(l) = 7 cm

∴ Lateral surface area (LSA) of the cube = 4l²

= 4 x 7² [putting value]

= 4 x 49

=196 cm²

To Read more....