Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Mensuration → Surface Areas and Volumes ↓


Volume of a Cube Cuboid and Cylinder


Measurement is a branch of mathematics that deals with the measurement of various geometric shapes and their parameters such as length, volume, area, etc. In this lesson, we are going to explore the concept of volume, particularly focusing on three-dimensional shapes: cube, cuboid, and cylinder, which are common in Class 10 mathematical studies.

Understanding volume

The concept of volume refers to the amount of space occupied by a three-dimensional object. Volume is measured in cubic units. For example, if the volume of an object is one cubic meter, it means that it can occupy a unit cube of dimensions exactly 1 meter x 1 meter x 1 meter.

Volume of a cube

A cube is a three-dimensional figure that has six equal square faces. All the edges of a cube are of equal length. The formula for finding the volume of a cube is given as:

Volume of a cube = side × side × side
V = a³

where a is the length of one side of the cube.

Let's imagine a cube:

If the length of one side of the cube is 5 cm, then the volume of the cube can be found as follows:

V = 5 cm × 5 cm × 5 cm = 125 cm³

Volume of a cuboid

A cuboid is a three-dimensional figure that has six rectangular faces, and the opposite faces are equal. The dimensions of a cuboid are its length, width, and height. The formula to find the volume of a cuboid is:

Volume of a cuboid = length × breadth × height
V = l × w × h

where l, w and h are the length, width and height of the cuboid respectively.

Let's imagine a cuboid:

If the dimensions of a cuboid are 8 cm, 5 cm and 3 cm, then the volume of the cuboid can be found as follows:

V = 8 cm × 5 cm × 3 cm = 120 cm³

Volume of a cylinder

A cylinder is a three-dimensional object consisting of two parallel circular bases connected by a curved surface. To find the volume of a cylinder we need to know the radius of its base and its height. The formula used for this is:

Volume of cylinder = π × radius² × height
V = πr²h

where r is the radius of the circular base and h is the height of the cylinder. The symbol π (pi) is a mathematical constant approximately equal to 3.14159.

Let's imagine a cylinder:

If the radius of a cylinder is 3 cm and height is 10 cm, then the volume of the cylinder can be found as follows:

V = π × (3 cm)² × 10 cm = π × 9 cm² × 10 cm
   = 90π cm³

About,

V ≈ 3.14159 × 90 cm³ ≈ 282.74 cm³

Applications of volume calculation

The ability to calculate the volume of these shapes is important for a variety of real-life scenarios, such as:

  • Determining how much space is available or needed for storage in cube containers.
  • Analyzing fluid dynamics and capabilities in engineering applications.
  • In architecture it is used to estimate material requirements based on quantity.

Summary

In conclusion, understanding the volume of various geometric shapes is important in both academic and practical applications. Each formula to calculate volume – whether it is for a cube, cuboid or cylinder – serves its own specific function. Mastering these formulas helps in solving complex problems and developing critical thinking skills required in higher studies.

It is essential to practice these concepts by solving many problems, starting from the basic level and gradually moving towards complex ones. Remember, practice is the key to mastering mathematical calculations!

Keep this guide as a reference whenever you need to review or clarify concepts related to the volume of a cube, cuboid, and cylinder.


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Volume of a Cube Cuboid and Cylinder

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