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9.5.2 - Cube

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Understanding the Cube

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Teacher
Teacher

Today, we will learn about the cube. A cube is actually a specific type of cuboid where all its sides are equal. Can anyone tell me how many faces a cube has?

Student 1
Student 1

It has six faces!

Teacher
Teacher

Correct! And what shape are those faces?

Student 2
Student 2

They are all squares.

Teacher
Teacher

Exactly! Now, can anyone think of a real-life example of a cube?

Student 3
Student 3

Dice are cubes.

Teacher
Teacher

Great example! Now let’s move on to calculate its volume. What do we use to find the volume of a cube?

Student 4
Student 4

We use V = l³, where l is the length of a side.

Teacher
Teacher

Absolutely! To remember this, think of the phrase 'Length Length Length' when calculating volume.

Calculating Volume of a Cube

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Teacher
Teacher

Now, let’s work on calculating the volume of cubes. If we have a cube with each side measuring 4 cm, what is its volume?

Student 1
Student 1

We would do 4 cm x 4 cm x 4 cm, which is 64 cm³.

Teacher
Teacher

Great job! And what if the side was 1.5 m instead?

Student 3
Student 3

Then it would be 1.5 x 1.5 x 1.5, which equals 3.375 m³.

Teacher
Teacher

Correct! Remember, when calculating volume, you can also think of it as taking the unit cube and stacking them up according to the formula.

Application and Importance of Volume

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Teacher
Teacher

Let’s think about why we calculate volume. Why might it be important to know how much space a cube occupies?

Student 4
Student 4

It could be used for packing materials.

Teacher
Teacher

Exactly! It's also crucial in industries where space is limited, such as shipping and storage. How about in construction? Why is knowing the volume of materials like concrete important?

Student 2
Student 2

Because we need the right amount to fill the space!

Teacher
Teacher

Very well said! Understanding the volume also helps in budgeting how much material is needed.

Introduction & Overview

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Quick Overview

This section presents the definition and volume formula of a cube, explaining its significance as a special type of cuboid.

Standard

In this section, we learn that a cube is a specific kind of cuboid with equal sides, and its volume can be calculated using the formula V = l³, where l is the length of a side. This formula highlights the significance of cubes in measuring volume in a three-dimensional space.

Detailed

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Audio Book

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Volume of Cube

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The cube is a special case of a cuboid, where l = b = h. Hence, volume of cube = l × l × l = l³.

Detailed Explanation

A cube is a three-dimensional shape with all sides of equal length. The formula for calculating the volume of a cube is derived from multiplying the length of one side (l) by itself three times. Since all sides are equal, we denote the length of a side as 'l'. Therefore, the volume (V) of the cube can be expressed with the formula V = l × l × l, which simplifies to V = l³. This means you can find the volume of a cube by measuring one side and then raising that measurement to the third power.

Examples & Analogies

Think of a cube as a box of chocolates, where each chocolate is exactly the same size and shape. If you know the length of one side of the box (for example, 4 cm), you can figure out how many chocolates can fit inside by calculating the volume. If each side is 4 cm, then the volume is 4 cm × 4 cm × 4 cm, which equals 64 cubic centimeters. This is the total space inside the box that can hold chocolates.

Try These Exercises

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TRY THESE Find the volume of the following cubes (a) with a side 4 cm (b) with a side 1.5 m.

Detailed Explanation

In the exercises provided, you're asked to calculate the volume of two different cubes based on their side lengths. For the first cube with a side of 4 cm, the computation would be: Volume = 4 cm × 4 cm × 4 cm = 64 cm³. For the second cube with a side of 1.5 m, you would first convert the measurement into centimeters for uniformity since 1 m equals 100 cm, leading to a side length of 150 cm. Consequently, Volume = 150 cm × 150 cm × 150 cm = 3,375,000 cm³, which also equals 3.375 m³ when converted back to cubic meters.

Examples & Analogies

Imagine you have two different boxes where you want to store stuffed animals. The first box is shaped like a cube, measuring 4 cm on each side. To find out how many stuffed animals can fit inside the box, you calculate the volume! The bigger box, at 1.5 m per side, can hold far more stuffed animals. This comparison shows how the volume increases dramatically with size.

Definitions & Key Concepts

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

Key Concepts

  • Cube: A three-dimensional shape with all equal sides.

  • Volume: Measure of how much space a solid occupies, calculated as l³ for cubes.

  • Cuboid: A more general term for three-dimensional rectangular shapes, which includes cubes.

Examples & Real-Life Applications

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

Examples

  • Example: Calculate the volume of a cube with a side of 4 cm. Solution: 4 × 4 × 4 = 64 cm³.

  • Example: Calculate the volume of a cube with a side of 1.5 m. Solution: 1.5 × 1.5 × 1.5 = 3.375 m³.

Memory Aids

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

🎵 Rhymes Time

  • A cube's volume is easy, you know, just take the side and let it grow. Length times length times length is the flow!

📖 Fascinating Stories

  • Imagine stacking blocks—a cube is like a box, where every side fits snugly together, just as all the blocks fit perfectly.

🧠 Other Memory Gems

  • Remember 'Cubic Length' to calculate the volume of a cube: Length × Length × Length.

🎯 Super Acronyms

CUBE = Calculating Under Basic Equations (used for volume V = l³).

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: Volume

    Definition:

    The amount of three-dimensional space an object occupies.

  • Term: Cuboid

    Definition:

    A three-dimensional shape with rectangular faces.