Core Area Theorems and Shortcut Techniques for Exams
A Brief Overview of Chapter 9 Areas of Parallelograms and Triangles
We know about geometry from the previous chapters where you have learned the properties of triangles and quadrilaterals. Now we will find out how to calculate surface areas of parallelograms and triangles by applying our knowledge of their properties. It will help you to understand how knowledge of geometry can be applied to solve real-life problems.
For instance, the formula for area of a rectangle can be used to find out the area of a large rectangular field. Additionally, a fundamental knowledge of class 9 areas of parallelogram and triangles are also used by engineers and architects while designing and constructing buildings.
CBSE Class 9 Maths Areas of Parallelograms and Triangles
You may know that a section of a plane bounded within a simple closed figure is called planar region and the measure of this region is known as its area. Apart from this, it would help if you kept in mind while studying areas of parallelograms and triangles that congruent figures or figures which have the same shape and size also have equal areas. However, two figures having the same area may not be congruent.
Thus, an area of a figure may be defined as a number in units that are associated with the planar region of the same. This definition has been discussed in detail in our NCERT solutions for class 9th maths chapter 9 areas of parallelograms and triangles.
You have learnt in previous classes the properties and formulae to calculate the area of various geometric figures like squares, rhombus, and rectangles. In this section, you will learn how to calculate areas of parallelograms and triangles lying on the same base and within the same parallels by applying that knowledge.
So, when are two figures said to be on the same base?
According to NCERT solutions class 9 maths chapter areas of parallelograms and triangles, two figures are on the same base and within the same parallels, if they have the following properties –
A Common base or side
Common vertices or vertex opposite to the common base and lying on a line which is parallel to the base.
Important Theorems
Given below are some theorems from 9th CBSE maths areas of parallelograms and triangles. It is based on the relation between two parallelograms lying on the same base and between the same parallels.
A thorough understanding of these theorems will enable you to solve subsequent exercises easily. You can cross-check your answers with our areas of parallelograms and triangles class 9 questions with answers.
Theorem 1:
Parallelograms on the same base and between the same parallels are equal in area. So,
A rectangle which is also a parallelogram lying on the same base and between same parallels also have the same area.
Hence the area of a parallelogram = base x height.
You can revise your answers with our areas of parallelograms and triangles class 9 exercise 9.2 solutions after attempting the questions on your own.
Theorem 2:
Two triangles which have the same bases and are within the same parallels have equal area. therefore -
Area of a triangle is ½ x base x height. You can practise questions in this theorem from areas of parallelograms and triangles exercise 9.3.
If a triangle and parallelogram are on the same base and between the same parallels, then the area of the triangle is equal to half the area of a parallelogram.
According to areas of parallelograms and triangles, Area of trapezium = ½ x (sum of parallel side) x (distance between them)
Area of a rhombus = ½ x product of the diagonals
Practise questions based on the theorem on your own and then check your answers with our areas of parallelograms and triangles class 9 exercise 9.3 solutions.
Theorem 3:
Triangles which have the same areas and lies on the same base, have their corresponding altitudes equal.
You can go through NCERT solutions for class 9th maths chapter 9 areas of parallelograms and triangles to gain more clarity on this theorem.
Our study materials on topics like areas of parallelograms and triangles are quite engaging and it aids students to learn and memorise important theorems and concepts easily.
Students can also sign up for our online interactive classes for doubt clearing and to know more about the topics such as areas of parallelograms and triangles answers. Now you can also download our Vedantu app for enhanced access.
FAQs on Areas of Parallelograms and Triangles Explained
1. What are the fundamental properties of a parallelogram relevant to its area?
The properties of a parallelogram crucial for understanding its area are:
Opposite sides are equal in length and parallel to each other.
A diagonal divides the parallelogram into two congruent triangles. Since congruent triangles have equal areas, the diagonal effectively splits the parallelogram's area in half.
Parallelograms on the same base and between the same parallels have the same area, regardless of their specific shape.
2. How is the formula for the area of a triangle (½ × base × height) derived from a parallelogram?
The formula for a triangle's area is directly derived from a parallelogram's area. If you take any triangle, you can construct a parallelogram that shares the same base and height. The diagonal of this parallelogram is the third side of the original triangle, dividing the parallelogram into two triangles of equal area. Therefore, the area of the triangle is precisely half the area of the parallelogram. Since the area of the parallelogram is base × height, the area of the triangle becomes ½ × base × height.
3. What are the main theorems covered in the chapter 'Areas of Parallelograms and Triangles'?
The key theorems in this chapter establish relationships between the areas of different geometric figures:
Theorem 1: Parallelograms that are on the same base and between the same parallel lines are equal in area.
Theorem 2: If a parallelogram and a triangle are on the same base and between the same parallels, the area of the triangle is half the area of the parallelogram.
Theorem 3: Triangles that are on the same base and between the same parallel lines are equal in area.
Theorem 4: A median of a triangle divides it into two triangles of equal areas.
4. What is the difference between an altitude and a median of a triangle?
While both are lines drawn from a vertex to the opposite side, they have different definitions and purposes:
An altitude is the perpendicular line segment from a vertex to its opposite side (or the extension of the opposite side). It represents the height of the triangle from that base and is used to calculate the area.
A median is a line segment that connects a vertex to the midpoint of the opposite side. Its primary property related to this chapter is that it divides the triangle into two smaller triangles of equal area.
5. Is the chapter 'Areas of Parallelograms and Triangles' in the CBSE Class 9 Maths syllabus for 2025-26?
No, according to the rationalised syllabus effective from the 2023-24 session onwards, the entire Chapter 9, 'Areas of Parallelograms and Triangles', has been removed from the CBSE Class 9 Mathematics curriculum for the academic year 2025-26. Students are not required to study this chapter for their final examinations.
6. Why are two triangles on the same base and between the same parallels equal in area?
Two triangles on the same base and between the same parallels have the same area because they share the same base length and the same height. The distance between two parallel lines is constant, so the altitude (height) of both triangles is identical. Since the area of any triangle is calculated as ½ × base × height, and both the base and height are the same for these two triangles, their areas must be equal.
7. How does a median of a triangle prove that it divides the triangle into two triangles of equal areas?
A median divides a triangle into two smaller triangles with equal areas. For example, in triangle ABC, if AD is the median to side BC, then D is the midpoint of BC, meaning BD = DC. Now, consider the two new triangles, ABD and ADC. They share the same vertex A, so their height (the altitude from A to BC) is the same. Since their bases are equal (BD = DC) and their heights are identical, their areas (½ × base × height) must be equal. Thus, ar(ABD) = ar(ADC).
8. What is the difference between congruent figures and figures with equal area?
This is a key conceptual distinction:
Congruent figures are identical in both shape and size. You can superimpose one perfectly over the other. All congruent figures will have equal areas.
Figures with equal area only have the same amount of enclosed space; their shapes can be completely different. For example, a rectangle of 6x4 (area 24 sq. units) and a triangle with base 8 and height 6 (area 24 sq. units) have equal areas but are not congruent. While congruence implies equal area, the reverse is not always true.
