Class 9 Mathematic Area of a Parallelogram and Triangles All Formula & Concept Notes | Formula Sheet

Here’s a detailed overview of the formulas and concepts related to the introduction of areas of parallelograms and triangles, which are typically covered in Class 9 mathematics:

1. Area of a Parallelogram

Concept:

• A parallelogram is a quadrilateral with opposite sides parallel and equal in length.
• The height (or altitude) of a parallelogram is the perpendicular distance between the two parallel sides.

Formula: $\text{Area} = \text{base} \times \text{height}$

• Base (b): Any one of the sides of the parallelogram.
• Height (h): The perpendicular distance from the chosen base to the opposite side.

Example: If a parallelogram has a base of 10 cm and a height of 5 cm, its area is: $\text{Area} = 10 \, \text{cm} \times 5 \, \text{cm} = 50 \, \text{cm}^2$

2. Area of a Triangle

Concept:

• A triangle is a three-sided polygon.
• The height of a triangle is the perpendicular distance from the chosen base to the opposite vertex.

Formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

• Base (b): Any one of the sides of the triangle.
• Height (h): The perpendicular distance from the base to the opposite vertex.

Example: If a triangle has a base of 8 cm and a height of 6 cm, its area is: $\text{Area} = \frac{1}{2} \times 8 \, \text{cm} \times 6 \, \text{cm} = 24 \, \text{cm}^2$

3. Relationship Between Parallelograms and Triangles

Concept:

• A triangle can be seen as a special case of a parallelogram. Specifically, if you take a parallelogram and draw one of its diagonals, you will create two congruent triangles.
• The area of a parallelogram is twice the area of one of these triangles.

Formula Derivation:

• The area of a parallelogram can be divided into two congruent triangles by drawing a diagonal.
• Therefore, if the area of the parallelogram is $A$ and the area of one of the triangles formed is $\frac{A}{2}$, then: $\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ $\text{Area of the parallelogram} = 2 \times \left(\frac{1}{2} \times \text{base} \times \text{height}\right)$

4. Applications and Examples

Example 1: Find the area of a parallelogram with a base of 12 cm and a height of 7 cm.

$\text{Area} = 12 \, \text{cm} \times 7 \, \text{cm} = 84 \, \text{cm}^2$

Example 2: Find the area of a triangle with a base of 9 cm and a height of 4 cm.

$\text{Area} = \frac{1}{2} \times 9 \, \text{cm} \times 4 \, \text{cm} = 18 \, \text{cm}^2$

Example 3: Given a parallelogram and one of its diagonals divides it into two triangles. If the diagonal is 10 cm long and the height from this diagonal to one of the sides is 5 cm, find the area of one of these triangles.

$\text{Area of one triangle} = \frac{1}{2} \times 10 \, \text{cm} \times 5 \, \text{cm} = 25 \, \text{cm}^2$

I hope this helps with understanding the introduction to areas of parallelograms and triangles! If you need more examples or further clarification.