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9.4.2 - Cube

Interactive Audio Lesson

Listen to a student-teacher conversation explaining the topic in a relatable way.

Understanding the Cube

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Teacher
Teacher

Today, we are going to explore a very interesting shape called a cube. Can anyone tell me what they know about cubes?

Student 1
Student 1

A cube has six faces, and they are all squares!

Student 2
Student 2

Also, all the sides are equal in length, right?

Teacher
Teacher

Exactly! That's a key property of a cube. Because all the sides are equal, we can use this to find the surface area. Can anyone remind me what we call the length of one side of the cube?

Student 3
Student 3

It's normally denoted by 'l' for length.

Teacher
Teacher

Correct! To find the area of one face, we will use the formula for the area of a square, which is side squared, or l². Since there are six faces, the total surface area formula will be 6l².

Student 4
Student 4

So if each side is 2 units long, the surface area would be 6 times 2²?

Teacher
Teacher

Exactly! Let's summarize this point: The surface area of a cube is calculated by 6 times the area of one face.

Finding Surface Area

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Teacher
Teacher

Now let's see how to apply the formula. If we have a cube with a side length of 3 cm, what would the total surface area be?

Student 1
Student 1

Using the formula, it would be 6 times 3², which is 6 times 9!

Student 2
Student 2

So the total surface area would be 54 cm²?

Teacher
Teacher

Spot on! This is the beauty of cubes and their uniformity. Now, let's look at a new challenge. If two cubes with side length 'b' are combined to form a cuboid, what do you think is the new surface area?

Student 3
Student 3

Would the surface area simply be the sum of their individual surface areas?

Teacher
Teacher

Good question! It's not that straightforward. We'll explore that in our next exercise.

In-depth Applications

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Teacher
Teacher

Let's tackle a practical problem. After painting a cube, we cut it into 64 smaller cubes. How many of these smaller cubes have no faces painted?

Student 4
Student 4

If the cube was initially 4x4x4, the inner cubes, which are not painted, would form another cube that is 2x2x2.

Student 1
Student 1

That means 2³, which equals 8 smaller cubes with no paint!

Teacher
Teacher

Excellent! And how about the number of cubes with only one face painted?

Student 2
Student 2

The cubes in the center of each face! There are 6 faces, and the center of each face has 1 cube, so 6 cubes!

Teacher
Teacher

Great job! The visualization of cubes and their surfaces really helps us understand three-dimensional geometry.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section discusses the properties and calculations related to the surface area of a cube.

Standard

The section covers how to determine the surface area of a cube, demonstrating it through hands-on activities and theoretical questions. It emphasizes the unique characteristic of a cube where all sides are of equal length, leading to a specific formula for the total surface area.

Detailed

In this section, we explore the cube, a three-dimensional shape characterized by having six equal square faces. The teaching begins with a hands-on activity where students create a net of a cube using squared paper and assemble it to visualize the shape better. The discussion covers important properties of the cube, such as its dimensions—length, width, and height being equal—and the area of each face. The section concludes with a formula for calculating the total surface area, highlighting the significance of understanding the cube's geometry in real-world contexts.

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Audio Book

Dive deep into the subject with an immersive audiobook experience.

Introduction to Cube

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DO THIS
Draw the pattern shown on a squared paper and cut it out [Fig 9.20(i)]. (You know that this pattern is a net of a cube. Fold it along the lines [Fig 9.20(ii)] and tape the edges to form a cube [Fig 9.20(iii)].

Detailed Explanation

This activity involves creating a physical model of a cube using its net. A net is a two-dimensional representation of a 3D shape, which can be folded to form that shape. When you draw the net of a cube, you will notice that it consists of 6 square faces. When these squares are cut out and taped together, they create the cube, demonstrating how 3D objects can be constructed from 2D patterns.

Examples & Analogies

Think of wrapping a gift. You lay out the wrapping paper (the net) flat and then fold it around the gift (the cube) to cover it completely. This shows how the 2D net becomes a 3D object.

Properties of Cube

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(a) What is the length, width and height of the cube? Observe that all the faces of a cube are square in shape. This makes length, height and width of a cube equal (Fig 9.21(i)).

Detailed Explanation

In a cube, all edges are of equal length, which means its length, width, and height are all the same. This property makes calculations easier since the area of each face will be the same, and it establishes that the cube is a regular polyhedron.

Examples & Analogies

Consider a dice. Each side of a dice is a square and has the same dimensions. This means if you know one side's length, you can easily determine the size of the entire cube.

Calculating Surface Area of Cube

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(b) Write the area of each of the faces. Are they equal?
(c) Write the total surface area of this cube.
(d) If each side of the cube is l, what will be the area of each face? (Fig 9.21(ii)).
Can we say that the total surface area of a cube of side l is 6l² ?

Detailed Explanation

Each face of the cube has an area that can be calculated by taking the side length (l) and squaring it (l²). Since a cube has 6 faces and all are identical, the total surface area can be found by multiplying the area of one face by 6. Therefore, the total surface area of a cube is given by the formula 6l².

Examples & Analogies

Imagine painting a large cardboard box (a cube). If you were to paint each side completely, you'd have to paint 6 identical surfaces. Knowing the size of one side allows you to calculate the total paint needed for all six sides.

Understanding Surface Areas through Examples

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TRY THESE
Find the surface area of cube A and lateral surface area of cube B (Fig 9.22).

Detailed Explanation

In this exercise, students are asked to apply what they've learned about the surface area of a cube. To find the total surface area, they will need to use the previously established formula for surface area. Lateral surface area, which refers to the sides of the cube without the top and bottom, can be calculated separately and can be useful in real-world applications.

Examples & Analogies

Think of a box of cereal; when painting the sides (lateral surface area) without covering the top or bottom, you need to consider only four out of the six sides, applying the same principles of surface area calculation in practical scenarios.

Discussion and Further Exploration

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THINK, DISCUSS AND WRITE
(i) Two cubes each with side b are joined to form a cuboid (Fig 9.23). What is the surface area of this cuboid? Is it 12b²? Is the surface area of cuboid formed by joining three such cubes, 18b²? Why?

Detailed Explanation

This discussion encourages students to think critically about how combining shapes affects their surface area. When two cubes are joined to form a cuboid, certain faces are not exposed anymore, reducing the total surface area compared to the sum of the individual cubes' faces. An analysis of this allows for understanding the geometric properties of shapes when combined.

Examples & Analogies

Imagine building with LEGO blocks. If two cubes are stacked, the surface area you can see (and thus paint or decorate) is less than if you had two separate cubes lying side by side because their bottom faces are touching and hidden from view.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Cube: A solid shape with six equal square faces.

  • Surface Area Formula: The total surface area can be represented as 6l² where l is the side length.

  • Nets: The net of a cube helps visualize how the surfaces come together.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example: A cube with a side length of 2 cm will have a surface area of 6*(2²) = 24 cm².

  • Example: Joining two cubes each with side length b results in a cuboid, requiring calculation of new surface areas.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • In a cube that's neat and tight, six square faces bring delight!

📖 Fascinating Stories

  • Imagine a box filled with tiny cubes. Each tiny cube can paint six sides, just like how a big cube's surface area equals six times the area of one face.

🧠 Other Memory Gems

  • To remember the surface area formula: 'Six Little Squares' for 6l²!

🎯 Super Acronyms

CUBE

  • C: for Corner
  • U: for Uniform size
  • B: for Box with squares
  • E: for Equals all sides.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Cube

    Definition:

    A three-dimensional shape with six equal square faces.

  • Term: Surface Area

    Definition:

    The total area of all the surfaces of a three-dimensional object.

  • Term: Net of a Cube

    Definition:

    A two-dimensional representation of the cube that can be folded to form it.