How to Find the Area of a Parallelogram and a Triangle Using Base and Height
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 with Formulas and Examples
1. What is the formula for the area of a parallelogram?
The area of a parallelogram is calculated using the formula Area = base × height.
- The base (b) is any one side of the parallelogram.
- The height (h) is the perpendicular distance from the base to the opposite side.
- So, A = b × h.
2. How do you find the area of a triangle?
The area of a triangle is given by the formula Area = 1/2 × base × height.
- Choose a side as the base (b).
- Measure the perpendicular height (h) from the opposite vertex.
- Apply A = 1/2 × b × h.
3. Why is the area of a triangle half the area of a parallelogram?
The area of a triangle is half the area of a parallelogram because a triangle can be formed by cutting a parallelogram along a diagonal.
- A parallelogram has area b × h.
- A diagonal divides it into two equal triangles.
- Each triangle therefore has area 1/2 × b × h.
4. What is the height of a parallelogram?
The height of a parallelogram is the perpendicular distance from the base to the opposite side.
- It is always drawn at a 90° angle to the base.
- It may lie inside or outside the parallelogram.
- It is not the slanted side unless it is perpendicular.
5. Can you find the area of a parallelogram without the height?
Yes, you can find the area of a parallelogram without the height if you know two sides and the included angle using Area = ab sinθ.
- a and b are adjacent sides.
- θ is the angle between them.
6. How do you calculate the area of a right triangle?
The area of a right triangle is calculated using Area = 1/2 × base × height, where the base and height are the two perpendicular sides.
- Identify the two sides forming the 90° angle.
- Multiply them together.
- Divide by 2.
7. What is the difference between the area of a parallelogram and a triangle?
The key difference is that a parallelogram uses Area = b × h while a triangle uses Area = 1/2 × b × h.
- A triangle’s area is always half the area of a parallelogram with the same base and height.
- Both formulas require a perpendicular height.
8. How do you find the area of a triangle using Heron’s formula?
The area of a triangle using Heron’s formula is Area = √[s(s − a)(s − b)(s − c)], where s is the semi-perimeter.
- First find s = (a + b + c)/2.
- Substitute into the formula.
9. What are the units for the area of parallelograms and triangles?
The units of area for parallelograms and triangles are always square units such as cm², m², or in².
- Area measures two-dimensional space.
- If lengths are in centimeters, area is in square centimeters (cm²).
10. What are common mistakes when finding the area of a parallelogram or triangle?
Common mistakes when calculating area include using the wrong height or forgetting the 1/2 factor for triangles.
- Using a slanted side instead of the perpendicular height.
- Forgetting the 1/2 in the triangle area formula.
- Mixing different measurement units.
- Not writing the answer in square units.
