6.2.1 - Cuboid
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Introduction to Cuboids
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Today, we're going to learn about cuboids. Can someone tell me what a cuboid is?
Is it like a box or a rectangular prism?
Exactly! A cuboid is a three-dimensional figure with six rectangular faces. What do you think makes up its dimensions?
Length, breadth, and height?
Right! Length, breadth, and height are crucial as they affect both surface area and volume. Let’s remember this using the acronym 'LBH'.
What do those letters stand for?
'L' for Length, 'B' for Breadth, and 'H' for Height. A handy way to remember!
So, what are some applications of cuboids, do you think?
They can be used in packaging and construction!
Exactly! Cuboids help us understand space and materials in real-life scenarios. Let's summarize: A cuboid is defined by its length, breadth, and height, and is applied in various practical contexts.
Calculating Surface Area
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Now let's talk about how we can calculate the surface area of a cuboid. Who can remind us of the formula?
It's 2 times (length times breadth plus breadth times height plus height times length)!
Perfect! We can simplify it using the surface area formula: $$ SA = 2(lb + bh + hl) $$ . Can anyone explain what each section of the formula means?
Each part of that formula accounts for the area of each pair of opposite sides.
Correct! If a cuboid has dimensions, say length 5 cm, breadth 3 cm, and height 4 cm, how would we calculate its surface area?
I would plug in the numbers! So, it would be: 2(5*3 + 3*4 + 4*5).
Right again! Can anyone calculate that?
It would be 2(15 + 12 + 20), so 2 times 47 gives us 94 cm².
Well done! Remember, surface area tells us how much space the surface of our cuboid occupies. Great work, class!
Calculating Volume
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We've covered surface area; now let's dive into volume! What's the formula for the volume of a cuboid?
Volume is length times breadth times height, right?
Exactly! The formula is $$ V = l imes b imes h $$. Can someone give me an example using specific numbers?
If we have a cuboid with length 5 cm, breadth 2 cm, and height 3 cm, the volume would be 5 times 2 times 3.
Correct! So, what’s the total volume?
That’s 30 cm³!
Right! Volume is crucial for understanding how much space is inside the cuboid. Remember to visualize it as filling a box with water. Let's recap: The volume formula is $$ V = l imes b imes h $$.
Introduction & Overview
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Quick Overview
Standard
In this section, we delve into the dimensions and properties of cuboids. Students learn how to calculate the surface area and volume using specific formulas, gaining an understanding of the relationship between the dimensions of length, breadth, and height.
Detailed
Cuboid
A cuboid is a three-dimensional geometrical figure bounded by six rectangular faces, wherein opposite faces are equal. The key aspects of studying cuboids in mensuration relate to calculating their surface area and volume.
Formulas:
- Surface Area (SA): The total area of all faces of the cuboid is calculated using the formula:
$$ SA = 2(lb + bh + hl) $$
where: l= Lengthb= Breadth-
h= Height -
Lateral Surface Area (LSA): The area of the vertical faces excluding the top and bottom is found using:
$$ LSA = 2h(l + b) $$ -
Volume (V): The measure of space within the cuboid is computed as:
$$ V = l imes b imes h $$
Understanding the properties and formulas related to cuboids plays an essential role in various practical applications, including construction and packaging.
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Surface Area of a Cuboid
Chapter 1 of 3
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Chapter Content
● Surface Area = 2(lb + bh + hl)
Detailed Explanation
The surface area of a cuboid is the total area of all its six rectangular faces. You can calculate it using the formula: Surface Area = 2(lb + bh + hl), where 'l' is the length, 'b' is the breadth, and 'h' is the height of the cuboid. This equation shows that you multiply the sum of the products of the different pairs of dimensions by 2, because each pair contributes two faces to the outer area.
Examples & Analogies
Imagine you are wrapping a rectangular box with gift paper. To cover the entire box, you would need to know how much paper is needed to cover all six sides. This is similar to calculating the surface area of a cuboid.
Lateral Surface Area of a Cuboid
Chapter 2 of 3
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Chapter Content
● Lateral Surface Area (LSA) = 2h(l + b)
Detailed Explanation
The lateral surface area of a cuboid refers to the area of the sides or the vertical faces, excluding the top and bottom bases. The formula to find the LSA is: LSA = 2h(l + b). Here, you are multiplying the height (h) by the sum of the length (l) and breadth (b), and then multiplying by 2 as there are two opposite sides for each dimension in a cuboid.
Examples & Analogies
Think of a soda can, which is shaped like a cylinder but has a similar concept. If you only want to paint the sides of the can without touching the top and bottom, you would use a similar method to calculate the surface area of the sides.
Volume of a Cuboid
Chapter 3 of 3
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Chapter Content
● Volume = l × b × h
Detailed Explanation
The volume of a cuboid is the space it occupies, which can be represented as: Volume = l × b × h. In this formula, you multiply the length (l) by the breadth (b) and the height (h) together. This calculation gives you the total cubic space inside the cuboid.
Examples & Analogies
Imagine filling a box with cereal. To know how much cereal you can fit inside, you calculate the volume of the box. If you know the dimensions (length, breadth, and height), you can easily find out how much space is available using the volume formula.
Key Concepts
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Cuboid: A 3D shape defined by its length, breadth, and height.
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Surface Area: The total area of all six rectangular faces of a cuboid.
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Volume: The space occupied within a cuboid, calculated as length × breadth × height.
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Lateral Surface Area (LSA): The sum of the areas of the sides (excluding top and bottom faces).
Examples & Applications
Example 1: Find the surface area of a cuboid with length 4 cm, breadth 3 cm, and height 2 cm.
Example 2: A cuboid has dimensions of 5 cm in height, 10 cm in length, and 3 cm in breadth. What is its volume?
Memory Aids
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Rhymes
Cuboid so neat, with edges so straight, Length, breadth, height, make all calculations great!
Stories
Imagine building a toy box in the shape of a cuboid, where knowing its volume helps you decide how many toys can fit inside. That is the story of a cuboid!
Memory Tools
Use 'LBH' to recall: Length, Breadth, Height for a cuboid's call!
Acronyms
Remember 'VLS' - Volume = Length x Height x Surface area!
Flash Cards
Glossary
- Cuboid
A three-dimensional geometric figure with six rectangular faces.
- Surface Area
The total area covering the surface of a three-dimensional object.
- Volume
The amount of space occupied by a three-dimensional object.
- Lateral Surface Area (LSA)
The total area of the sides (excluding top and bottom) of a three-dimensional object.
- Dimensions
Measurements that define the size of the object, typically length, breadth, and height.
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