Step-by-Step Guide to Calculating Volume of Shapes
The volume of an object or a closed surface is a mathematical quantity that shows how often three-dimensional space it occupies. The volume is measured in cubic units such as m3, cm3, and so on.
Volume is sometimes spoken to as capacity. The volume of a cylindrical jar, for example, is used to calculate how much water it can contain.
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Volume of Cuboid
A cuboid is a solid geometrical object with 6 faces.
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Volume of the cuboid(V) = length × breadth × height
Volume of Cube
A cube is a solid geometrical object with 6 faces.
All the sides of the cube are equal in length.
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Volume of the cube(V) = length × breadth × height
Volume of the cube(V) = s × s × s (where s is the side of the cube)
Volume of the cube(V) = s3
Volume by Counting Unit Cubes
We know that Volume is defined as a space occupied by a three-dimensional figure.
Let’s understand the concept of counting unit cubes with the help of the following examples:
Example: Find the volume of the given figure. Take the volume of each small cube as 1cm3.
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Solution
Step 1: We have to number the cubes.
A total of 6 cubes are present in the given figure.
So, cubes are numbered from 1 to 6.
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Step 2: Calculate the number of layers in the given figure.
A total of 2 layers are present in the given figure.
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Step3: Calculate the volume of each layer
Volume of layer= (Number of cubes in the layer × volume of small cube)
Volume of layer 1 = 3 × 1cm3 = 3 cm3
Volume of layer 2 = 3 × 1cm3 = 3 cm3
Step4 : Calculate the Total volume
Total volume= Volume of layer 1 + Volume of layer 2
So, Total Volume of figure = 3 cm3 + 3 cm3
Total Volume of figure = 6 cm3.
Solved Questions
1. Find the volume of the given figure. Take the volume of each small cube as 1cm3.
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Ans: Layer 1 contains 6 cubes,
So, Volume of layer 1 = 6 × 1cm3 = 6 cm3
Layer 2 contains 6 cubes,
So, Volume of layer 2 = 6 × 1cm3 = 6 cm3
Layer 3 contains 12 cubes, (6 in front + 6 in back)
So, Volume of layer 3 = 12 × 1cm3 = 12 cm3
So, Total Volume of figure = 6 cm3+6 cm3+12 cm3
Total Volume of figure = 24 cm3.
2. Find the volume of the cube having side 3 cm.
Ans: Volume of cube(V) = s3
Volume of cube(V) = (3)3
Volume of cube(V) = 3 cm × 3 cm × 3 cm
Volume of cube(V) = 27 cm3
3.Find the volume of the cuboid having l = 6 cm, b = 4 cm and h = 5 cm.
Ans: Volume of cuboid(V) = length × breadth × height
Volume of cuboid(V) = 6 cm × 4 cm × 5 cm
Volume of cuboid(V) = 120 cm3.
4. What is the volume of the pictures given below. Take the volume of each small cube as 1cm3.
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Ans: For figure A:
There is a total of 5 cubes
So, the Volume of figure A = 5 × 1cm3 = 5 cm3
For figure B:
There are a total of 12 cubes
(6 cubes in layer 1+ 6 cubes in layer 2)
So, Volume of figure B = 12 × 1cm3 = 12 cm3
Fun Facts:
When you link two points with only a line segment, you get a one-dimensional object that can only be measured in length.
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Two-dimensional figures are flat and have two dimensions i.e length and width. The area of a two-dimensional figure can be calculated by using length and width.
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The objects we come across on a daily basis are solid, three-dimensional objects with the following dimensions: length, width, and depth. The volume for three-dimensional objects is used to estimate their size.
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Summary
In this article we have discussed the concept of volume. First, we discussed volume definition, the volume of cube and cuboid concept and formulas, fun facts, and finally solved the problems. We have learned how to find the volume of figures by counting the unit cubes.
Learning by Doing
1. Jerry was flying a kite that was yellow in shade. Suddenly, a powerful wind blew, and his kite became tangled in a huge tree. Jerry-built a ladder made of cubical boxes. Now let us count the volume of the ladder to help Jerry in getting his yellow kite. Take the volume of each small cube as 1cm3.
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2. Tim wants to pack his old books in the cubical box. Help Tim in finding the volume of the cubical box on the side 5 cm.
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3. Miss Mary went shopping and bought too many clothes. Now, Miss Mary wants to place all her clothes in her cuboidal cupboard. Let's help Miss Mary in finding the volume of the cuboidal cupboard whose l = 4 cm, b = 3 cm, and h = 6 cm.
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FAQs on Volume of Geometrical Figures Explained
1. What is the volume of a geometrical figure?
The volume of a geometrical figure is the amount of three-dimensional space it occupies. It measures the total space enclosed within the boundaries of a 3D object. For instance, the volume of a box tells you how much sand, water, or air it can hold.
2. What is the difference between area and volume?
The key difference lies in the dimensions they measure. Area is the measure of space on a flat, two-dimensional (2D) surface, like the surface of a sheet of paper, and is measured in square units (e.g., cm²). In contrast, volume is the measure of space occupied by a three-dimensional (3D) object, like a ball or a box, and is measured in cubic units (e.g., cm³).
3. Why is volume measured in cubic units like cm³ or m³?
Volume is measured in cubic units because it is calculated by multiplying three dimensions: length, width, and height. Each dimension has a unit (like cm). When you multiply them (cm × cm × cm), the result is a cubic unit (cm³). This unit represents a tiny cube with sides of that specific length, and the volume tells you how many of these tiny cubes are needed to fill the entire shape.
4. What are the formulas for calculating the volume of common 3D shapes?
The formulas for the volume of common geometrical figures are essential for calculations. Key examples include:
- Cube: Volume = side × side × side = s³
- Cuboid: Volume = length × width × height = l × w × h
- Cylinder: Volume = π × radius² × height = πr²h
- Sphere: Volume = (4/3) × π × radius³ = (4/3)πr³
- Cone: Volume = (1/3) × π × radius² × height = (1/3)πr²h
5. How is the volume of a regular geometrical figure calculated in practice?
To calculate the volume of a regular geometrical figure, you follow a simple process. First, you identify the shape (e.g., a cube, cylinder, or sphere). Next, you measure its required dimensions (like side length, radius, or height). Finally, you substitute these measurements into the specific volume formula for that shape to find the result.
6. What is the difference between volume and capacity?
While closely related, volume and capacity are not the same. Volume is the total space an object occupies, including the material it's made of. Capacity, on the other hand, refers to the amount a hollow object can hold inside it. For example, the capacity of a bottle is the amount of liquid it can contain, whereas its volume would also include the glass itself. Capacity is often measured in litres (L) or millilitres (mL).
7. What does the volume of a figure actually tell you?
The volume of a figure provides crucial information about its size in three dimensions. It tells you the quantity of material needed to make a solid object (e.g., how much concrete is in a pillar) or the holding capacity of a container (e.g., how much water a swimming pool can hold). In essence, it quantifies the 'bigness' of an object in 3D space.
8. Can you give some real-life examples where understanding volume is important?
Understanding volume is essential in many real-life situations. For example:
- Cooking and Baking: Measuring ingredients like milk or flour using cups or spoons is a direct application of volume.
- Construction: Calculating the amount of concrete needed for a foundation or sand for a sandbox requires volume calculations.
- Shipping and Logistics: The cost of shipping packages is often based on their dimensional weight, which is derived from volume.
- Medicine: Dosages of liquid medicines are prescribed in specific volumes, like 5 mL.
9. How does changing the dimensions of a shape, like a cuboid, affect its volume?
The volume of a cuboid is directly proportional to each of its dimensions (length, width, and height). This means if you double just one dimension (e.g., the height), the total volume will also double. However, if you double all three dimensions, the new volume will be 2 × 2 × 2 = 8 times the original volume. This shows that changes in dimensions have a significant, multiplicative impact on an object's volume.
10. Do irregular shapes have a volume, and how can it be measured?
Yes, all 3D objects, including irregular ones like a stone or a potato, have a volume. For such shapes where a formula cannot be applied, we can use the water displacement method. This involves submerging the object in a container of water (like a measuring cylinder) and measuring the volume of water it pushes aside. The volume of the displaced water is equal to the volume of the irregular object.
