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What is the Volume of a Combined Solid?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The volume of a combined solid is the total space occupied by an object made up of two or more simpler solids joined together. To find it, you calculate the volume of each individual solid and then add them up.
Simple Example
Quick Example
Imagine you have a toy truck made by joining a cuboid (the body) and a cylinder (the fuel tank). To know the total space the truck takes up, you would find the volume of the cuboid part and the volume of the cylinder part separately, then add those volumes together.
Worked Example
Step-by-Step
Let's find the volume of a pencil sharpener shaped like a cuboid with a small triangular prism on top.
Step 1: Identify the shapes. It's a cuboid and a triangular prism.
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Step 2: Find the dimensions of the cuboid. Let its length be 5 cm, width be 2 cm, and height be 1 cm.
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Step 3: Calculate the volume of the cuboid. Volume of cuboid = length x width x height = 5 cm x 2 cm x 1 cm = 10 cubic cm.
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Step 4: Find the dimensions of the triangular prism. Let the base of the triangle be 2 cm, its height be 1 cm, and the length of the prism be 5 cm (same as cuboid length).
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Step 5: Calculate the volume of the triangular prism. Volume of triangular prism = (1/2 x base x height of triangle) x length of prism = (1/2 x 2 cm x 1 cm) x 5 cm = 1 cm^2 x 5 cm = 5 cubic cm.
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Step 6: Add the volumes of both shapes. Total Volume = Volume of cuboid + Volume of triangular prism = 10 cubic cm + 5 cubic cm = 15 cubic cm.
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Answer: The total volume of the pencil sharpener is 15 cubic cm.
Why It Matters
Understanding combined volumes is super important for engineers designing buildings or bridges, and for product designers creating new items. It helps in calculating how much material is needed, how much space an object will take, or how much liquid a container can hold. This skill is used in fields like architecture, manufacturing, and even in computer graphics for creating realistic 3D models.
Common Mistakes
MISTAKE: Forgetting to calculate the volume of ALL the individual parts of the combined solid. | CORRECTION: Break down the combined solid into ALL its basic shapes first, then calculate each volume.
MISTAKE: Using the wrong formula for one of the basic shapes (e.g., using area formula instead of volume, or using the wrong volume formula). | CORRECTION: Double-check the volume formulas for each basic 3D shape (cuboid, cube, cylinder, cone, sphere, prism).
MISTAKE: Adding up the dimensions (like lengths) instead of the calculated volumes. | CORRECTION: Remember that volume is about space, not just length. Always calculate the individual volumes first, then add those volumes.
Practice Questions
Try It Yourself
QUESTION: A water tank is shaped like a cylinder with a hemisphere on top. If the cylinder has a radius of 3 meters and a height of 7 meters, and the hemisphere has the same radius, what is the total volume of the tank? (Use pi = 22/7) | ANSWER: Volume of cylinder = pi * r^2 * h = (22/7) * 3 * 3 * 7 = 198 cubic meters. Volume of hemisphere = (2/3) * pi * r^3 = (2/3) * (22/7) * 3 * 3 * 3 = 18 * (22/7) = 396/7 = 56.57 cubic meters (approx). Total Volume = 198 + 56.57 = 254.57 cubic meters (approx).
QUESTION: A building is shaped like a cuboid with a pyramid on its roof. The cuboid base is 10 meters by 8 meters, and its height is 6 meters. The pyramid has the same base dimensions (10m x 8m) and a height of 3 meters. What is the total volume of the building? | ANSWER: Volume of cuboid = 10 * 8 * 6 = 480 cubic meters. Volume of pyramid = (1/3) * base area * height = (1/3) * (10 * 8) * 3 = (1/3) * 80 * 3 = 80 cubic meters. Total Volume = 480 + 80 = 560 cubic meters.
QUESTION: Imagine a 'gola' (ice candy) stand with an ice block shaped like a large cube (side 20 cm) with a smaller cuboid (10 cm x 10 cm x 5 cm) carved out from its center top. What is the remaining volume of the ice block? | ANSWER: Volume of large cube = 20 * 20 * 20 = 8000 cubic cm. Volume of carved-out cuboid = 10 * 10 * 5 = 500 cubic cm. Remaining Volume = 8000 - 500 = 7500 cubic cm.
MCQ
Quick Quiz
A toy is made of a cone placed on top of a cylinder. If the cone has a volume of 50 cubic cm and the cylinder has a volume of 120 cubic cm, what is the total volume of the toy?
70 cubic cm
170 cubic cm
240 cubic cm
6000 cubic cm
The Correct Answer Is:
B
To find the total volume of a combined solid, you simply add the volumes of its individual parts. So, 50 cubic cm (cone) + 120 cubic cm (cylinder) = 170 cubic cm.
Real World Connection
In the Real World
When ISRO launches rockets, these rockets are often made of different sections (cylinders, cones) joined together. Engineers calculate the volume of each part to determine fuel capacity, the weight of the rocket, and how much space it will occupy. Even architects designing a new school building need to calculate the total volume of classrooms, hallways, and offices to estimate air conditioning needs and construction material quantities.
Key Vocabulary
Key Terms
Volume: The amount of three-dimensional space an object occupies | Combined Solid: An object made by joining two or more basic 3D shapes | Cuboid: A 3D shape with six rectangular faces | Cylinder: A 3D shape with two circular bases and a curved surface | Hemisphere: Half of a sphere
What's Next
What to Learn Next
Great job learning about combined volumes! Next, you can explore the 'Surface Area of Combined Solids'. This will teach you how to calculate the total outer surface area of these complex shapes, which is important for painting or wrapping objects.
