9.5.2 - Cube
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Understanding the Cube
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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?
It has six faces!
Correct! And what shape are those faces?
They are all squares.
Exactly! Now, can anyone think of a real-life example of a cube?
Dice are cubes.
Great example! Now let’s move on to calculate its volume. What do we use to find the volume of a cube?
We use V = l³, where l is the length of a side.
Absolutely! To remember this, think of the phrase 'Length Length Length' when calculating volume.
Calculating Volume of a Cube
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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?
We would do 4 cm x 4 cm x 4 cm, which is 64 cm³.
Great job! And what if the side was 1.5 m instead?
Then it would be 1.5 x 1.5 x 1.5, which equals 3.375 m³.
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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Let’s think about why we calculate volume. Why might it be important to know how much space a cube occupies?
It could be used for packing materials.
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?
Because we need the right amount to fill the space!
Very well said! Understanding the volume also helps in budgeting how much material is needed.
Introduction & Overview
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Quick Overview
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
Cube
A cube is a three-dimensional geometric figure that has all its sides of equal length. Each face of a cube is a square, and because of this unique property, a cube can be viewed as a specific type of cuboid. In this section, we discuss how to calculate the volume of a cube, denoted by the formula V = l³, where "l" represents the length of one side. This formula is pivotal in learning how to understand and apply concepts of volume measurement for solid shapes in practical contexts. Calculating the volume of a cube is straightforward—simply multiplying the length of one side by itself twice (l × l × l). The importance of understanding how to compute the volume of a cube extends to various real-life applications, such as in architecture, packaging, and many areas of science and engineering.
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Volume of Cube
Chapter 1 of 2
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Chapter Content
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
Chapter 2 of 2
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Chapter Content
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.
Key Concepts
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Cube: A three-dimensional shape with all equal sides.
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Volume: Measure of how much space a solid occupies, calculated as l³ for cubes.
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Cuboid: A more general term for three-dimensional rectangular shapes, which includes cubes.
Examples & Applications
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
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Rhymes
A cube's volume is easy, you know, just take the side and let it grow. Length times length times length is the flow!
Stories
Imagine stacking blocks—a cube is like a box, where every side fits snugly together, just as all the blocks fit perfectly.
Memory Tools
Remember 'Cubic Length' to calculate the volume of a cube: Length × Length × Length.
Acronyms
CUBE = Calculating Under Basic Equations (used for volume V = l³).
Flash Cards
Glossary
- Cube
A three-dimensional shape with six equal square faces.
- Volume
The amount of three-dimensional space an object occupies.
- Cuboid
A three-dimensional shape with rectangular faces.
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