Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Surface Areas and Volumes ↓


Volume of a Cube Cuboid and Cylinder


The volume of 3D shapes like cubes, cuboids, and cylinders are essential concepts to understand the capacity or space these shapes can hold. When we talk about volume, we are referring to the amount of space occupied by a shape in three dimensions. Let us look at each of these geometric shapes in detail and learn how to calculate their volume.

Understanding volume

Volume is a measure of the space occupied by a solid object. It is usually measured in cubic units, such as cubic centimeters (cm 3), cubic meters (m 3), etc. Think of volume as how many unit-sized cubes can fit inside the shape.

Volume of a cube

A cube is a special type of cuboid, with all sides of equal length. A cube has six square faces, twelve edges, and eight corners. A cube is identified by the length of its edge.

Formula for volume of a cube

The volume V of a cube with side length a is given by the formula:

V = a 3

This simply means that you multiply the length of one side by itself twice.

Example calculation

If the side of a cube is 3 cm, then its volume is:

V = 3 cm x 3 cm x 3 cm = 27 cm 3

Thus, the volume of the cube is 27 cubic centimeters.

Visual example

Volume of a cuboid

A cuboid, also called a rectangular prism, has six rectangular faces. It is identified by its length, width, and height.

Formula for the volume of a cuboid

The volume V of a cuboid with length l, width w and height h is given by the formula:

V = lxwxh

This formula simply multiplies the length, width, and height of the cuboid.

Example calculation

If the length of a cuboid is 4 cm, width 3 cm and height 2 cm, then its volume is:

V = 4 cm x 3 cm x 2 cm = 24 cm 3

This means that the cuboid can occupy 24 cubic centimeters of space.

Visual example

Volume of a cylinder

A cylinder is a solid object with two parallel circular bases connected by a curved surface. A cylinder is identified by the radius of its base and its height.

Formula for the volume of a cylinder

The volume V of a cylinder with base radius r and height h is given by the formula:

V = πr 2 h

Here, π (pi) is approximately 3.14159. This formula calculates the area of the base circle and multiplies it by the height.

Example calculation

If the radius of the base of a cylinder is 5 cm and height is 10 cm, then its volume is:

V = π x (5 cm) 2 x 10 cm = π x 25 cm 2 x 10 cm = 250π cm 3

When π is estimated the result is approximately 785.398 cm 3.

Visual example

Applications of volume

Understanding the volume of a cube, cuboid, and cylinder is very practical in our daily lives. From figuring out how much water a tank can hold to estimating the volume of packages, volume calculations help in decision making. These calculations are used in engineering, architecture, medicine, and other fields.

Lesson example - Cube

Consider a small toy box that is shaped like a cube with a side of 10 cm. To find out how much space is inside, find its volume:

V = 10 cm x 10 cm x 10 cm = 1000 cm 3

This means that the capacity of the toy box is 1000 cubic centimeters.

Lesson example - Cuboid

Suppose you have a wooden plank in the shape of a cuboid with dimensions 2m by 0.5m by 0.1m. Calculate its volume:

V = 2m x 0.5mx 0.1m = 0.1m 3

Therefore, the plank occupies 0.1 cubic meters of space.

Text example - Cylinder

If the base radius of an oil barrel is 0.7m and height 2m, calculate how much oil can be stored in it:

V = π x (0.7m) 2 x 2m = π x 0.49m 2 x 2m = 0.98π m 3

When π is estimated the barrel can hold approximately 3.08 cubic meters of weight.

Conclusion

Finding the volume of a cube, cuboid, and cylinder helps us understand and measure the space inside these objects. Using the formulas, you can solve a variety of practical problems involving these shapes. Remember, volume is a straightforward concept that simply involves multiplying dimensions or using a formula to see how much space an object has.


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Surface Area of a Cube Cuboid and Cylinder
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Surface Area and Volume of a Cone

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Volume of a Cube Cuboid and Cylinder

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