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 ↓


Areas of Parallelograms and Triangles


Understanding the area of shapes like parallelograms and triangles is an important aspect of geometry. This concept is not only important in the academic world, but it also has practical applications in fields like architecture, engineering, and design.

Parallelogram

A parallelogram is a four-sided shape in which opposite sides are parallel and equal in length. The most basic examples of parallelograms include squares, rectangles, and rhombuses.

Properties of parallelogram

  • The opposite sides are equal and parallel.
  • Opposite angles are equal.
  • The diagonals of a parallelogram bisect each other.

Formula for the area of a parallelogram

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

Area = base × height

Here, the base is any side of the parallelogram, and the height is the perpendicular distance from the base to the opposite side.

Let's imagine a parallelogram:

Base Height

Consider a parallelogram whose base = 8 cm and height = 5 cm. Using the formula:

Area = 8 cm × 5 cm = 40 cm 2

Triangle

A triangle is a three-sided polygon with three angles. There are different types of triangles based on the side length and angle measure, such as equilateral, isosceles, scalene, acute-angled, obtuse-angled, and right-angled triangles.

Properties of triangles

  • The sum of the interior angles of a triangle is always 180 degrees.
  • The exterior angle of a triangle is equal to the sum of its opposite interior angles.
  • The area of a triangle depends on its base and height.

Triangle area formula

The area of a triangle is calculated using the following formula:

Area = (1/2) × base × height

The base can be any one side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex.

Let's imagine a triangle:

Base Height

For example, if the base of a triangle is 10 cm and the height is 6 cm, then its area can be found as follows:

Area = (1/2) × 10 cm × 6 cm = 30 cm 2

Comparative example

Parallelogram vs triangle

Since the formulas for both parallelograms and triangles use the base and height, it is useful to compare them:

  • The area of a parallelogram with base 8 cm and height 5 cm is 40 cm 2.
  • The area of a triangle with equal base and height is (1/2) × 8 cm × 5 cm = 20 cm2.

Common mistakes and tips

  • Always make sure the height is perpendicular to the base.
  • The overall height should be measured from the base line, not from a vertex to a base.
  • Double-check that you're using the correct unit of measurement.

By consistently practicing these calculations and visualizing the shapes and dimensions involved, you can develop a strong understanding of how to effectively determine the area of both parallelograms and triangles.

These principles will prove useful not only in academic pursuits, but also in the real world, where these patterns and formulas are often used in various businesses.

Remember, the art of geometry is not just about memorizing formulas; it is about understanding the shapes and dimensions that make up the world around us.


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Areas of Parallelograms and Triangles

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