Learn
Games

12.3 - Volume of a Combination of Solids

Interactive Audio Lesson

Listen to a student-teacher conversation explaining the topic in a relatable way.

Understanding Volume of Combined Solids

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

Teacher
Teacher

Today, we will explore how to find the volume of solids that are combined from basic shapes. Can anyone tell me what they think volume means?

Student 1
Student 1

Isn't it the amount of space an object occupies?

Teacher
Teacher

Exactly! And when we have a solid that is made up of other solids, we just add their volumes together. Does everyone understand that?

Student 2
Student 2

But if we join two shapes, do we measure the surface they join?

Teacher
Teacher

Great question! Unlike surface areas, we actually sum the total volumes of the contributing solids since no part of the volume disappears. Let's look at an example.

Example Calculation of a Shed

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

Teacher
Teacher

Imagine a shed shaped like a cuboid with a half-cylinder on top. First, how do we find the volume of the cuboid?

Student 3
Student 3

We multiply the length, width, and height!

Teacher
Teacher

Correct! And for the half-cylinder, we use the formula for a cylinder but divide the result by two. Together, those give us the total volume of the shed.

Student 4
Student 4

What happens if there are things in the shed, like machines?

Teacher
Teacher

Good point! We would subtract the volume occupied by those items to find out the volume of air left in the shed. Now, let’s work through the calculation.

Real-World Application: A Glass with a Hemisphere

Unlock Audio Lesson

Signup and Enroll to the course for listening the Audio Lesson

Teacher
Teacher

Let’s consider a glass that has a cylindrical body and a hemispherical bottom. How would we find its total capacity?

Student 1
Student 1

We find the volume of the cylinder and the hemisphere separately and then add them?

Teacher
Teacher

Exactly right! By calculating the volumes separately, we can then find the actual capacity of the glass. This teaches us about real-life applications of these concepts.

Student 2
Student 2

What if we wanted to know how much syrup it could hold?

Teacher
Teacher

Then we would ensure that we account for the shape when determining how much liquid it can really hold. You’re all grasping the concepts quite well!

Introduction & Overview

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

Quick Overview

This section focuses on calculating the volumes of solids formed by combining two or more basic solids.

Standard

In this section, we explore the concept of how to compute the volume of combined solids such as a cuboid with a hemisphere or a cone. Unlike surface areas, when solids are joined, their volumes sum up directly to provide the total volume. Practical examples illustrate this concept clearly.

Detailed

Volume of a Combination of Solids

In this section, we learn how to calculate the volume of solids that are formed by joining two or more basic shapes, like cones, cylinders, cuboids, and hemispheres. It’s significant to remember that, unlike surface areas where some sections may not be counted (for instance, the joined surfaces), the total volume is simply the sum of the volumes of the individual solids. We examine various examples, such as a shed shaped like a block with a half-cylinder on top and a glass with a hemispherical bottom, to illustrate the principle of combining volumes effectively. Each example serves not only to highlight the computational aspects but also to represent real-world situations for better understanding.

Youtube Videos

Problem types: surface area of combination of solids | Class 10 (India) | Math
Problem types: surface area of combination of solids | Class 10 (India) | Math
Surface Areas And Volumes | Chapter 12 | Introduction |
Surface Areas And Volumes | Chapter 12 | Introduction |
Surface Area of Combination of Solids
Surface Area of Combination of Solids
surface area and volume class 10 all formulas || #SHORTS
surface area and volume class 10 all formulas || #SHORTS
Surface area of combination of solids | Surface area and volume | Class 10 (India) | Math
Surface area of combination of solids | Surface area and volume | Class 10 (India) | Math
Surface Areas of Combination of Solids - Surface Areas and Volumes | Class 10 Maths Chapter 12 |CBSE
Surface Areas of Combination of Solids - Surface Areas and Volumes | Class 10 Maths Chapter 12 |CBSE
Class 10- Surface Areas and Volumes 🔥 - 1 Shot Animation - Maths Chapter 12
Class 10- Surface Areas and Volumes 🔥 - 1 Shot Animation - Maths Chapter 12
Circles, Surface Areas & Volumes | Areas Related to Circles | Class-10th | One Shot Revision
Circles, Surface Areas & Volumes | Areas Related to Circles | Class-10th | One Shot Revision
Class 10th Surface Areas and Volumes One Shot 🔥 | Class 10 Maths Chapter 12 | Shobhit Nirwan
Class 10th Surface Areas and Volumes One Shot 🔥 | Class 10 Maths Chapter 12 | Shobhit Nirwan
CBSE Class 10 Maths | Pair of Linear Equations in Two Variables | Surface Areas & Volumes
CBSE Class 10 Maths | Pair of Linear Equations in Two Variables | Surface Areas & Volumes

Audio Book

Dive deep into the subject with an immersive audiobook experience.

Introduction to Volume Calculations

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

In the previous section, we have discussed how to find the surface area of solids made up of a combination of two basic solids. Here, we shall see how to calculate their volumes.

Detailed Explanation

This chunk introduces the concept of calculating the volumes of combined solids. While previously we learned how to find the surface areas of solids made from basic shapes (like spheres or cylinders), here we focus on how to combine their volumes. It emphasizes that when calculating volume, we don’t lose any part of the volume as we did with surface areas—when two solids join, we simply add their volumes together.

Examples & Analogies

Think about filling up a jar with different types of marbles. If you have a red marble (say a sphere) and a green marble (a cube), the total amount of space taken up by both marbles in the jar is simply the sum of the spaces they occupy individually.

Example 5: Volume of a Shed

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

Example 5: Shanta runs an industry in a shed which is in the shape of a cuboid surmounted by a half cylinder (see Fig. 12.12). If the base of the shed is of dimension 7 m × 15 m, and the height of the cuboidal portion is 8 m, find the volume of air that the shed can hold...

Detailed Explanation

This example outlines a practical scenario where you calculate the volume of air in a shed shaped like a cuboid with a half cylinder on top. The volume is calculated by adding the volume of the cuboidal part (length × breadth × height) to the volume of the half cylinder (which is half of the volume of a full cylinder). This illustrates the direct application of volume calculations for combined shapes, demonstrating that their total volume is simply the sum of each part’s volume.

Examples & Analogies

Imagine the shed as a large food storage container. If you know the area of the floor and the height of the walls, you can easily calculate how much food it can store. By adding the extra storage area provided by the half-cylinder top, you're maximizing the amount of food that can be safely kept in this container.

Example 6: Capacity of a Glass

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

Example 6: A juice seller was serving his customers using glasses as shown in Fig. 12.13. The inner diameter of the cylindrical glass was 5 cm, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass...

Detailed Explanation

In this example, we analyze a glass with a unique shape: a cylinder with a hemispherical base. The apparent capacity (the full volume the glass would hold if it were a simple cylinder) is calculated first, followed by subtracting the volume of the hemispherical portion since it takes up space inside the glass. This process illustrates the necessity of accounting for internal shapes while calculating volume.

Examples & Analogies

Think of the glass as a piggy bank that has a small dome at the bottom. If you want to know how much money you can fit inside, you must first figure out how much space the whole piggy bank has, and then subtract the space taken up by the dome at the bottom to determine how much cash you can actually deposit.

Example 7: Volume of a Toy

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

Example 7: A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm...

Detailed Explanation

This example walks through the process of calculating the volume of a toy shaped like a cone on top of a hemisphere. The volume of both shapes is calculated separately, utilizing the formulas for the volume of a cone and a hemisphere, and added together. This displays how different shapes can interact to define a larger object’s volume.

Examples & Analogies

Visualize a birthday hat that combines a pointed cone shape with a round base that looks like half of a ball. If you wanted to know how much space that hat or toy occupies, you would measure the total space taken by both the cone and the hemisphere, which is just as straightforward as stacking boxes on top of each other to determine their combined height.

Definitions & Key Concepts

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

Key Concepts

  • Volume of Combined Solids: The total volume is the sum of the volumes of the individual solids.

  • Surface vs Volume: The surface area may not include overlapping surfaces, but all the volumes are counted.

  • Practical Applications: Understanding combined volumes helps in real-life situations, such as construction and manufacturing.

Examples & Real-Life Applications

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

Examples

  • The volume of a shed made of a cuboid and a half-cylinder can be calculated by adding the two individual volumes.

  • A glass that is a combination of a cylinder and a hemisphere requires separate calculations of both parts to find its overall capacity.

Memory Aids

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

🎵 Rhymes Time

  • Volume in space, let’s sum it up, solids combined, fill our cup.

📖 Fascinating Stories

  • Imagine a factory with boxes (cuboids) and round containers (cylinders) stacking together. Every box adds its own space to the factory's total. Each time they combine, the total volume ticks up!

🧠 Other Memory Gems

  • C-V-C: Calculate Volume of Composites. (C for Combined, V for Volume).

🎯 Super Acronyms

S-V (Sum of Volumes)

  • To find the volume of combined solids
  • remember that 'S' is for sum and 'V' is for volume.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Volume

    Definition:

    The amount of space that a substance or object occupies.

  • Term: Cuboid

    Definition:

    A three-dimensional shape with six rectangular faces.

  • Term: Cylinder

    Definition:

    A solid shape with straight parallel sides and a circular or oval cross-section.

  • Term: Hemisphere

    Definition:

    Half of a sphere, divided along a great circle.

  • Term: Combined Solid

    Definition:

    A solid formed by the joining of two or more basic solids.