Area of Parallelogram and Triangle for Class 9

Geometry introduces us to the concepts of area calculations for parallelograms and triangles. Let's delve into the world of geometric wonders, understanding how to measure and make sense of the space enclosed by Parallelogram and Triangle.

A parallelogram has four sides with opposite sides parallel and equal in length. Calculating the area of a triangle involves a formula: Area = Base × Height

A Triangle has three sides and three angles. Calculating the area of a triangle involves a formula: Area = ½ × Base × Height

Planar Region

A planar region refers to the part of a flat surface that is enclosed by a closed figure. The measure of this region is the area of that figure and we express it using numbers along with a unit.

Polygonal Region

A polygonal region means the combination of polygons (like a triangle, quadrilateral, pentagon, etc.) and area inside it. Every polygonal region has an area.
For example, if we take a triangle and everything is within its borders, the whole region inside the triangle is called the triangular area.

Properties of the Area of a Figure

⇒ If figures are congruent, they have equal areas.

If two shapes have the same shape and size, we say they are congruent. And when two figures are congruent, it means their areas are also the same.
For example, if two triangles △ABC and △DEF are congruent, then the area of △ABC is equal to the area of △DEF.

⇒ Figures with equal areas may not necessarily be congruent.

If the two shapes have the same area, then it is not necessary that they are the same shape. Hence, they are not congruent.

Figures between the same parallels

The figures between the same parallels are used to compare the properties and areas of various shapes within this specific geometric arrangement.

Area Theorems

Area theorems are fundamental principles in geometry that provide rules and formulae for calculating the area enclosed by different shapes. Here are some key area theorems commonly used:

Theorems on Area of Triangle

Theorems on Area of Parallelogram

Here are some key points to understand about areas:

1. Parallelogram Area: The area of a parallelogram is the product of its sides and the height (altitude) corresponding to that side.
2. Figures Between Parallels: Figures are considered to be between the same parallels and on the same base if they share a common side and their opposite vertices lie on a line parallel to the common side.
3. Diagonals of a Parallelogram: A diagonal of a parallelogram divides it into two triangles of equal areas.
4. Equality of Parallelograms: Parallelograms on the same or equal base and between the same parallel lines are equal in area.
   Parallelograms that share the same or equal base and have equal areas are situated between the same parallel lines.
5. Rectangle-Parallelogram Equality: A rectangle and a parallelogram on the same base and between the same parallels have the same area.
6. Triangle Area: The area of a triangle is half the product of any side and the height (altitude) corresponding to that side.
7. Equality of Triangles: Triangles on the same or equal base and between the same parallel lines are equal in area.
   Triangles that share the same or equal base and have equal areas are situated between the same parallel lines.
8. Triangle-Parallelogram Relationship: If a triangle and a parallelogram share the same or equal base and are between the same parallel lines, then the area of the triangle is half that of the parallelogram.
9. Equal Areas and Altitudes: Triangles with equal areas and with one side of one triangle equal to a side of the other triangle have corresponding height (altitude) that are equal.
10. Median of a Triangle: A median of a triangle divides it into two triangles of equal area.
11. Trapezium Area: The area of a trapezium is half the product of the sum of the lengths of the parallel sides and its height.