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


Measurement is an important concept in mathematics that deals with the measurement of geometric shapes and their parameters such as length, area, and volume. In simple terms, it is the study of various formulas for finding various geometric quantities.

Understanding mensuration

The subject of measurement helps us measure everything around us. Imagine you are trying to paint a room, wrap a box, or fill a water tank. In each of these cases, you need to measure the measurement of the measurement. It is necessary to know how much paint is needed, how much wrapping paper is needed, or how much water will fill a tank. Measurement gives us the tools and formulas to find these quantities.

Basic concepts

The main concepts of measurement include perimeter, area, and volume:

  • The perimeter is the distance around a two-dimensional shape.
  • Area is a measure of the space enclosed within a two-dimensional shape.
  • Volume is the measure of the space occupied by a three-dimensional object.

Circumference

The perimeter is the total length of the boundary of a two-dimensional shape. Let's look at some basic shapes and their perimeters:

Perimeter of a rectangle

lB

The perimeter of a rectangle can be calculated by adding the lengths of the four sides. The formula is given as follows:

Perimeter of Rectangle = 2 × (Length + Breadth)

Perimeter of a square

A

A square is a special case where all sides are equal. Therefore, the perimeter of the square is:

Perimeter of Square = 4 × Side

Area

Area is the region within a shape. Let's find the area of some common shapes:

Area of a rectangle

The area of a rectangle can be calculated using the following formula:

Area of Rectangle = Length × Breadth

Area of a square

Since all sides of the square are equal, its area will be:

Area of Square = Side × Side = Side²

Area of a circle

R

A circle is a shape whose all points are at the same distance from its center. This distance is called the radius (r). The formula for the area of a circle is:

Area of Circle = π × r²

Volume

Volume measures the space occupied by a three-dimensional object. Here are some common volumes:

Volume of a cube

A cube has all its sides of equal length, and this can be found as follows:

Volume of Cube = Side × Side × Side = Side³

Volume of a cuboid

A cuboid is similar to a cube, but has different length, width, and height. Its volume is given by:

Volume of Cuboid = Length × Breadth × Height

Volume of a cylinder

A cylinder is a 3D object with a circular base and a fixed height. Its volume is calculated as follows:

Volume of Cylinder = π × r² × h

Examples and applications

Let's look at some practical examples of mensuration.

Example 1: Finding the area of a garden

Imagine you have a rectangular garden 20 meters long and 10 meters wide. To find the amount of grass or tile needed, calculate the area:

Length = 20m Breadth = 10m Area of Garden = Length * Breadth = 20m * 10m = 200 m²

Therefore, 200 square metres of grass is required to cover the garden.

Example 2: Painting a wall

Suppose you want to paint a square wall, each side of which is 5 m long. The area to be painted can be found as follows:

Side = 5m Area to be painted = Side² = 5m * 5m = 25 m²

You will need enough paint to cover an area of 25 square metres.

Example 3: Water to fill the tank

If you have a cylindrical tank with a radius of 1.5 m and a height of 3 m, you can calculate the amount of water needed to fill the tank:

Radius = 1.5m Height = 3m Volume = π * (1.5m)² * 3m = π * 2.25 m² * 3m = 21.205 m³ (approx., using π = 3.14159)

The tank will be able to hold approximately 21.205 cubic metres of water.

Conclusion

Measurement is an essential branch of mathematics that provides the tools we need to measure and quantify the world around us. From simple two-dimensional shapes to complex three-dimensional objects, measurement helps us find perimeter, area, and volume. helps. It is applicable in various real-life scenarios like construction, architecture and even simple daily tasks like packing and painting.

Don't forget to keep these formulas in mind:

  • Perimeter of rectangle = 2 × (length + breadth)
  • Area of a rectangle = length × breadth
  • Volume of cylinder = π × r² × h

Practice with different shapes and measurements to improve your understanding and application of measurement. With practice, these calculations will become second nature.


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