9.4.1 - Cuboid
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Introduction to Cuboids
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Today we are going to learn about cuboids! Can anyone tell me what a cuboid is?
Is it like a box with six rectangular faces?
Exactly! A cuboid has three pairs of identical rectangular faces. Now, does anyone know how we can calculate its total surface area?
I think we need the dimensions of the cuboid?
Correct! We need the length, width, and height. The formula is 2(lb + bh + hl). Remember it as 'LWH' for length, width, and height!
Can you explain the formula again?
Certainly! When we multiply the dimensions, we get the area of individual faces. We then multiply by 2 because there are two of each face.
That’s neat! So what do we do with the numbers?
Great question! Let’s practice using an example together.
Surface Area Calculations
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Let’s say we have a cuboid that measures 20 cm in height, 15 cm in length, and 10 cm in width. Who can calculate its total surface area?
I think we can plug the numbers into the formula!
Great idea! So, SA = 2(20 × 15 + 20 × 10 + 10 × 15). Can anyone calculate that?
Okay, let's see... 20 × 15 = 300, 20 × 10 = 200, and 10 × 15 = 150.
So adding those gives us 300 + 200 + 150 = 650, and then multiply by 2 to get 1300 cm²!
Exactly! You’ve got it! So the total surface area is 1300 cm². Remember to keep practicing with different numbers.
What's the lateral surface area?
Lateral surface area is calculated by 2h(l + b). So let's do that next.
Lateral Surface Area
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Now let’s calculate the lateral surface area of our cuboid. We have h = 20 cm, l = 15 cm, and b = 10 cm.
What’s the formula again?
The formula is 2h(l + b). So in this case, it would be 2 × 20 × (15 + 10).
Okay! That would be 2 × 20 × 25, which equals 1000 cm²!
Correct! The lateral surface area is 1000 cm². Great job! Can anyone remind me what the lateral surface area represents?
It’s the area of the walls of the cuboid, not counting the top and bottom!
Well done! Remember, understanding these calculations will help us in practical applications, like packaging.
Introduction to Cuboids
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Today we are going to learn about cuboids! Can anyone tell me what a cuboid is?
Is it like a box with six rectangular faces?
Exactly! A cuboid has three pairs of identical rectangular faces. Now, does anyone know how we can calculate its total surface area?
I think we need the dimensions of the cuboid?
Correct! We need the length, width, and height. The formula is 2(lb + bh + hl). Remember it as 'LWH' for length, width, and height!
Can you explain the formula again?
Certainly! When we multiply the dimensions, we get the area of individual faces. We then multiply by 2 because there are two of each face.
That’s neat! So what do we do with the numbers?
Great question! Let’s practice using an example together.
Surface Area Calculations
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Let’s say we have a cuboid that measures 20 cm in height, 15 cm in length, and 10 cm in width. Who can calculate its total surface area?
I think we can plug the numbers into the formula!
Great idea! So, SA = 2(20 × 15 + 20 × 10 + 10 × 15). Can anyone calculate that?
Okay, let's see... 20 × 15 = 300, 20 × 10 = 200, and 10 × 15 = 150.
So adding those gives us 300 + 200 + 150 = 650, and then multiply by 2 to get 1300 cm²!
Exactly! You’ve got it! So the total surface area is 1300 cm². Remember to keep practicing with different numbers.
What's the lateral surface area?
Lateral surface area is calculated by 2h(l + b). So let's do that next.
Lateral Surface Area
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Now let’s calculate the lateral surface area of our cuboid. We have h = 20 cm, l = 15 cm, and b = 10 cm.
What’s the formula again?
The formula is 2h(l + b). So in this case, it would be 2 × 20 × (15 + 10).
Okay! That would be 2 × 20 × 25, which equals 1000 cm²!
Correct! The lateral surface area is 1000 cm². Great job! Can anyone remind me what the lateral surface area represents?
It’s the area of the walls of the cuboid, not counting the top and bottom!
Well done! Remember, understanding these calculations will help us in practical applications, like packaging.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, we learn about the total surface area of a cuboid, derived formulae, and lateral surface area. We also see practical examples illustrating these concepts.
Detailed
Detailed Summary
In this section, we explore the concept of the cuboid, which is a three-dimensional geometric figure outlined by six rectangular faces. The total surface area of a cuboid is essential for various real-world applications, like packaging and construction. The formula for calculating the total surface area (SA) is:
Total Surface Area of a Cuboid:
SA = 2(lb + bh + hl)
Where:
- l = length
- b = breadth (width)
- h = height
This formula indicates that there are three pairs of identical rectangles in a cuboid. The area of each rectangular face needs to be calculated and summed to find the total surface area.
In addition, we also discuss lateral surface area (the area excluding the base and top) which is given by:
Lateral Surface Area = 2h(l + b).
We delve into a practical example: if a cuboid has dimensions of 20 cm (height), 15 cm (length), and 10 cm (width), the total surface area would be calculated as:
- SA = 2(20 × 15 + 20 × 10 + 10 × 15) = 1300 cm².
This section prepares the learners to understand and apply the concept of surface area to various cuboid-related problems.
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Audio Book
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Understanding the Cuboid
Chapter 1 of 4
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Chapter Content
Suppose you cut open a cuboidal box and lay it flat. We can see a net as shown below. Write the dimension of each side. You know that a cuboid has three pairs of identical faces. What expression can you use to find the area of each face?
Detailed Explanation
A cuboid is a three-dimensional shape that has six faces, which are rectangular. When you cut open a cuboidal box and lay it flat, you create a 'net'. The net shows how the faces of the cuboid are arranged and provides an understanding of the dimensions involved. Each pair of identical faces means that the area expression for one face can be reused for the other faces, simplifying the calculation.
Examples & Analogies
Think about a shoebox. If you were to take the shoebox apart and lay the cardboard flat, you’d see how each side fits together, just like a net. This helps you visualize how to calculate the area of each side.
Calculating Total Surface Area
Chapter 2 of 4
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Chapter Content
Find the total area of all the faces of the box. We see that the total surface area of a cuboid is area I + area II + area III + area IV + area V + area VI = h × l + b × l + b × h + l × h + b × h + l × b
Detailed Explanation
To find the total surface area of a cuboid, you sum the areas of all six faces. The expression includes all combinations of its height (h), length (l), and breadth (b). The formula simplifies this to: \( Total Surface Area = 2(lb + bh + hl) \). This means you calculate each area once and then double the sum, as each of the three dimensions has two corresponding faces.
Examples & Analogies
Imagine wrapping a gift in paper. You measure each of the sides of the cuboid-shaped gift (length, width, and height) to determine how much wrapping paper you need. By calculating each side's area and then summing them, you ensure you have enough paper to cover the entire gift.
Application of Surface Area Calculation
Chapter 3 of 4
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Chapter Content
Suppose the height, length and width of the box shown above are 20 cm, 15 cm and 10 cm respectively. Then the total surface area = 2 (20 × 15 + 20 × 10 + 10 × 15).
Detailed Explanation
By substituting the specific dimensions into the surface area formula, we can calculate the exact total surface area of the cuboid. This allows for real-world applications, such as determining how much paint is needed to cover the surface of a cuboidal box.
Examples & Analogies
When painting a room, knowing the surface area of the walls helps you figure out how many cans of paint to buy. Just like calculating the surface area of the cuboid helps to determine how much paint is needed, measuring the walls of a room helps you estimate your painting supplies.
Lateral Surface Area of a Cuboid
Chapter 4 of 4
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Chapter Content
The side walls (the faces excluding the top and bottom) make the lateral surface area of the cuboid. Hence, the lateral surface area of a cuboid is given by 2(h × l + b × h).
Detailed Explanation
The lateral surface area only accounts for the walls of the cuboid, ignoring the top and bottom. This is important for applications such as painting or covering sides while leaving the base exposed.
Examples & Analogies
Imagine painting only the sides of a cupboard while leaving the top and bottom unpainted. You would only measure the areas of the vertical faces, not the horizontal surfaces. This is exactly what is done when calculating the lateral surface area.
Key Concepts
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Total Surface Area: Calculation of all external surfaces of a cuboid.
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Lateral Surface Area: Area of sides only about the base and top.
-
Dimensions: Measurements that define the size of a cuboid.
Examples & Applications
Example 1: For a cuboid with length = 5 cm, width = 3 cm, and height = 4 cm, the total surface area is 2(53 + 34 + 5*4) = 94 cm².
Example 2: Lateral surface area for the same cuboid would be 24(5 + 3) = 64 cm².
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
For a cuboid tall and wide, add all six sides, it’s a ride!
Stories
Imagine a box storing secrets, each side needs painting; to know how much, we add all its faces!
Memory Tools
For 'SURFACE AREA', remember: 'Six' sides - 'Unique' pairs.
Acronyms
LWH - Length, Width, Height; use these for calculations right!
Flash Cards
Glossary
- Cuboid
A three-dimensional figure with six rectangular faces.
- Surface Area
The total area that the surface of an object occupies.
- Lateral Surface Area
The area of the sides of a three-dimensional object.
- Dimensions
Measurements of length, width, and height.
Reference links
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