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12.2 - Surface Area of a Combination of Solids

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Understanding Surface Areas

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Teacher
Teacher

Today, we are going to explore the surface area of solids that are combinations of basic shapes. Can anyone explain what we mean by 'surface area'?

Student 1
Student 1

Is it the total area that covers the outside of a 3D shape?

Teacher
Teacher

Exactly! Now, when we combine shapes, like a cylinder and a hemisphere, how might we approach calculating their total surface area?

Student 2
Student 2

We can break them down into their curved surfaces and add them together!

Teacher
Teacher

Great! We can think of the formula: TSA = CSA of cylinder + CSA of hemisphere. Remember that CSA stands for Curved Surface Area.

Student 3
Student 3

What if there are more solids involved, like a cone as well?

Teacher
Teacher

We would do the same, adding the relevant CSAs. Let’s summarize this understanding: when dealing with combinations of solids, analyze each shape individually.

Example Problems

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Teacher
Teacher

Let’s apply our earlier discussion with an example. Rasheed wants to color his toy shaped like a cone and hemispherical top. What steps should we take?

Student 4
Student 4

First, we need to find the curved surface areas of the hemisphere and cone.

Teacher
Teacher

Correct! The curved surface area of the hemisphere is given by the formula 2πr², where r is the radius.

Student 1
Student 1

And for the cone, it would be πrl, right?

Teacher
Teacher

Exactly! After calculating those, we add them to find the total surface area for coloring. Remember, we don't include the base of the hemisphere since it's attached.

Student 2
Student 2

So, is the total surface area just the sum of both?

Teacher
Teacher

Yes, just the CSAs. Keeping our shapes distinct is key in finding their total surface area.

Breaking Down Complex Shapes

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Teacher
Teacher

Now, let’s talk about a complex shape. How would we simplify it for surface area calculations?

Student 3
Student 3

By isolating each solid and finding their surface areas individually!

Teacher
Teacher

Correct! Think of a wooden toy rocket built from a cone and a cylinder. How would we approach that?

Student 4
Student 4

Calculate the CSA of the cone and the CSA of the cylinder and then add them.

Teacher
Teacher

Absolutely right! However, remember to adjust for overlapping areas where they're joined. This applies to many combinations, like a cylinder with a hemispherical depression, as well!

Student 1
Student 1

So this means the surface areas might not always add up as we expect?

Teacher
Teacher

Exactly! It’s essential to double-check which surfaces are exposed when solids are combined.

Introduction & Overview

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

Quick Overview

This section explains how to calculate the surface area of solids formed by combining basic geometric shapes.

Standard

In this section, we explore how to determine the surface area of complex solids made up of simple shapes such as cylinders, cones, and hemispheres. We demonstrate key concepts through examples and emphasize the importance of breaking down the shapes into manageable components.

Detailed

In this section, we delve into the concept of calculating the surface area of combinations of simple solids like cylinders, cones, and hemispheres. The approach to solving these problems involves breaking the complex solid into its individual components and calculating the relevant areas separately. For instance, when analyzing a container shaped like a cylinder with two hemispheres on either end, we can find the total surface area (TSA) by adding the curved surface area (CSA) of the cylinder and the CSAs of each hemisphere. The section includes practical examples, such as a toy top shaped like a cone and hemisphere, and provides a step-by-step solution to illustrate the application of these calculations. Additionally, we emphasize that the total surface area is not simply the sum of the individual surface areas due to overlapping sections where solids are combined.

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Audio Book

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Identifying the Solid

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Let us consider the container seen in Fig. 12.2. How do we find the surface area of such a solid? Now, whenever we come across a new problem, we first try to see, if we can break it down into smaller problems, we have earlier solved. We can see that this solid is made up of a cylinder with two hemispheres stuck at either end. It would look like what we have in Fig. 12.4, after we put the pieces all together.

Detailed Explanation

To find the surface area of a complex solid, we can dissect it into simpler shapes that we already know how to calculate the surface area for. In this example, we have a container that resembles a cylinder with two hemispherical ends. Recognizing the individual shapes allows us to apply our existing knowledge about calculating the surface area of a cylinder and a hemisphere.

Examples & Analogies

Think of it like building a model of a house using blocks. You wouldn't try to paint the whole house at once; instead, you can paint each block one at a time, applying the same principles of color to each section.

Calculating Total Surface Area

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If we consider the surface of the newly formed object, we would be able to see only the curved surfaces of the two hemispheres and the curved surface of the cylinder. So, the total surface area of the new solid is the sum of the curved surface areas of each of the individual parts. This gives,

TSA of new solid = CSA of one hemisphere + CSA of cylinder + CSA of other hemisphere
where TSA, CSA stand for ‘Total Surface Area’ and ‘Curved Surface Area’ respectively.

Detailed Explanation

The total surface area (TSA) of the new solid is calculated by summing the curved surface areas (CSA) of each distinct solid that makes it up. For our solid with cylindrical and hemispherical sections, we include only the curved surface areas because the flat surfaces where they join are not visible.

Examples & Analogies

Imagine you are wrapping a birthday gift with shiny wrapping paper. You would only cover the visible sides of the box and the bow on top, ignoring the parts that are already covered or stuck together.

Combining Hemisphere and Cone

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Let us now consider another situation. Suppose we are making a toy by putting together a hemisphere and a cone. Let us see the steps that we would be going through.

First, we would take a cone and a hemisphere and bring their flat faces together. Here, of course, we would take the base radius of the cone equal to the radius of the hemisphere, for the toy is to have a smooth surface. So, the steps would be as shown in Fig. 12.5.

Detailed Explanation

When combining a hemisphere and a cone, we align their flat faces to ensure they fit together smoothly. It is essential that the radius of the cone’s base matches that of the hemisphere so there are no gaps. This combination creates a solid that collectively has a new surface area calculated from both components.

Examples & Analogies

Think of making a smoothie. Just like blending different fruits together creates a new flavor, combining a cone and a hemisphere into one solid shape creates a new toy design!

Calculating Surface Area of the Toy

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Now if we want to find how much paint we would require to colour the surface of this toy, what would we need to know? We would need to know the surface area of the toy, which consists of the CSA of the hemisphere and the CSA of the cone.

So, we can say:
Total surface area of the toy = CSA of hemisphere + CSA of cone.

Detailed Explanation

To determine how much paint is needed for the toy, we first need to calculate its surface area. This requires knowing the curved surface area of the hemisphere and the cone. By summing these two areas, we find the total area on which paint will be applied.

Examples & Analogies

Imagine getting a canvas ready for painting. You first measure the area to know how much paint you need. The same concept applies here: we measure the surface area of our toy to know how much paint to buy.

Example Problem: Coloring the Top

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Example 1: Rasheed got a playing top (lattu) as his birthday present, which surprisingly had no colour on it. He wanted to colour it with his crayons. The top is shaped like a cone surmounted by a hemisphere (see Fig 12.6). The entire top is 5 cm in height and the diameter of the top is 3.5 cm. Find the area he has to colour. (Take π = 22/7)

Detailed Explanation

This example illustrates the method of applying the concepts discussed. Here, Rasheed's top combines a cone and a hemisphere, and we can determine the surface area he needs to color by using the known dimensions. We calculate the curve surfaces separately and sum them up, leveraging our understanding of the shapes involved.

Examples & Analogies

Think of Rasheed's top like making your own ice cream sundae. You have the cone for the ice cream and the flat bowl on top. When determining the cost of toppings, you first need to measure both components to make sure you have enough!

Definitions & Key Concepts

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

Key Concepts

  • Surface Area of Combination: The method of calculating the surface area for solids formed by combining geometrical shapes.

  • Breaking Down Solids: The importance of separating complex shapes into simpler components for easier calculation.

Examples & Real-Life Applications

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

Examples

  • Example 1: A toy shaped like a cone and a hemisphere. Calculate the total surface area using respective formulas for cone and hemisphere.

  • Example 2: A decorative block combining a cube and a hemisphere. Calculate total surface area, considering components correctly.

Memory Aids

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

🎵 Rhymes Time

  • In the cone's height and base, CSA is not far from grace!

📖 Fascinating Stories

  • Imagine a painter wanting to coat a toy rocket. He must measure the cone and cylinder, making sure not to cover the bottom where they meet.

🧠 Other Memory Gems

  • TSA = CSA + CSA for Cylinder and Cone needs to be known.

🎯 Super Acronyms

CST - Curved Surface Total; it's the mix of cones and any solid’s total!

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Surface Area

    Definition:

    The total area that the surface of a three-dimensional object occupies.

  • Term: Curved Surface Area (CSA)

    Definition:

    The area of the curved surface of 3D shapes excluding their bases.

  • Term: Total Surface Area (TSA)

    Definition:

    The sum of the areas of all the surfaces of a three-dimensional object.

  • Term: Cylinder

    Definition:

    A 3D shape with two parallel circular bases connected by a curved surface.

  • Term: Cone

    Definition:

    A 3D shape with a circular base tapering to a point called the apex.

  • Term: Hemisphere

    Definition:

    Half of a sphere, divided by a plane passing through its center.