NCERT Solutions for Exercise 12.1 Class 9 Maths Chapter 12 - Heron's Formula

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# NCERT Solutions for Exercise 12.1 Class 9 Maths Chapter 12 - Heron's Formula

Edited By Vishal kumar | Updated on Oct 11, 2023 08:09 AM IST

## NCERT Solutions for Class 9 Maths Chapter 12: Heron's Formula Exercise 12.1- Free PDF Download

NCERT Solutions for Class 9 Maths Chapter 12: Heron's Formula Exercise 12.1- Heron's Formula will be unlocked in 9th class maths exercise 12.1 answers. When you know the lengths of a triangle's sides, you can use this formula to get its area.

To begin, do class 9 maths chapter 12 exercise 12.1, which is similar to the first stage in this adventure. It tests you with tasks that require you to compute triangle areas using Heron's Formula. What's more, guess what? The solutions to these problems are available in PDF format for free. This means you can access them even when you're not connected to the internet, making your journey through triangles and Heron's Formula even more accessible.

NCERT Solutions for Class 9 Maths chapter 12 exercise 12.1 is about what the heron formula is and to use and apply heron formula in questions and problems. The basic idea of NCERT solutions for exercise 12.1 class 9 maths is that If the quadrilaterals are broken into triangular portions, Heron's formula may also be used to calculate the area of a quadrilateral. Apply Heron's formula to find the area of individual triangular portions once the quadrilaterals have been separated into triangular shapes. After you've calculated the areas of each triangle portion, add them all up to get the quadrilateral's area.

**As per the CBSE Syllabus for 2023-24, please note that this chapter has been renumbered as Chapter 10. So, when referring to this chapter, consider it as Chapter 10 - Heron's Formula.

## Access Heron's Formula C lass 9 Chapter 12 Exercise: 12.1

Given the perimeter of an equilateral triangle is 180cm.

So, $3a = 180\ cm$ or $a = 60\ cm$ .

Hence, the length of the side is 60cm.

Now,

Calculating the area of the signal board by the Heron's Formula:

$A = \sqrt{s(s-a)(s-b)(s-c)}$

Where, s is the half-perimeter of the triangle and a, b and c are the sides of the triangle.

Therefore,

$s = \frac{1}{2}Perimeter = \frac{1}{2}180cm = 90cm$

$a =b=c = 60cm$ as it is an equilateral triangle.

Substituting the values in the Heron's formula, we obtain

$\implies A = \sqrt{90(90-60)(90-60)(90-60)} = 900\sqrt{3}\ cm^2$ .

From the figure,

The sides of the triangle are:

$a = 122m,\ b = 120m\ and\ c = 22m$

The semi perimeter, s will be

$s = \frac{a+b+c}{2} = \frac{122+120+22}{2} = \frac{264}{2} = 132m$

Therefore, the area of the triangular side wall will be calculated by the Heron's Formula,

$A = \sqrt{s(s-a)(s-b)(s-c)}$

$= \sqrt{132(132-122)(132-120)(132-22)}\ m^2$

$= \sqrt{132(10)(12)(110)}\ m^2$

$= \sqrt{(12\times11)(10)(12)(11\times10)}\ m^2 = 1320\ m^2$

Given the rent for 1 year (i.e., 12 months) per meter square is Rs. 5000.

Rent for 3 months per meter square will be:

$Rs.\ 5000\times \frac{3}{12}$

Therefore, for 3 months for 1320 m 2 :

$Rs.\ 5000\times \frac{3}{12}\times 1320 = Rs.\ 16,50,000.$

Given the sides of the triangle are:

$a = 15 m,\ b= 11m\ and\ c= 6m.$

So, the semi perimeter of the triangle will be:

$s = \frac{a+b+c}{2} = \frac{15+11+6}{2} = \frac{32}{2} = 16m$

Therefore, Heron's formula will be:

$A = \sqrt{s(s-a)(s-b)(s-c)}$

$= \sqrt{16(16-15)(16-11)(16-6)}$

$= \sqrt{16(1)(5)(10)}$

$= \sqrt{(4\times 4)(1)(5)(5\times 2 )}$

$= 4\times 5 \sqrt2 = 20\sqrt2\ m^2$

Hence, the area painted in colour is $20\sqrt2\ m^2$ .

Given the perimeter of the triangle is $42cm$ and the sides length $a= 18cm$ and $b= 10cm$

So, $a+b+c = 42cm$

Or, $c = 42 - 18-10 = 14cm$

So, the semi perimeter of the triangle will be:

$s = \frac{P}{2} = \frac{42cm}{2} = 21cm$

Therefore, the area given by the Heron's Formula will be,

$A = \sqrt{s(s-a)(s-b)(s-c)}$

$= \sqrt{21(21-18)(21-10)(21-14)}$

$= \sqrt{(7\times3 )(3)(11)(7)}$

$= 21\sqrt{11}\ cm^2$

Hence, the area of the triangle is $21\sqrt{11}\ cm^2.$

Given the sides of a triangle are in the ratio of $12:17:25$ and its perimeter is $540cm$

Let us consider the length of one side of the triangle be $a = 12x$

Then, the remaining two sides are $b = 17x$ and $c = 25x.$

So, by the given perimeter, we can find the value of x:

$Perimeter = a+b+c = 12x+17x+25x = 540cm$

$\implies 54x = 540cm$

$\implies x = 10$

So, the sides of the triangle are:

$a = 12\times10 =120 cm$

$b = 17\times10 =170 cm$

$c = 25\times10 =250 cm$

So, the semi perimeter of the triangle is given by

$s = \frac{540cm}{2} = 270cm$

Therefore, using Heron's Formula, the area of the triangle is given by

$A = \sqrt{s(s-a)(s-b)(s-c)}$

$= \sqrt{270(270-120)(270-170)(270-250)}$

$= \sqrt{270(150)(100)(20)}$

$= \sqrt{81000000} = 9000cm^2$

Hence, the area of the triangle is $9000cm^2$ .

The perimeter of an isosceles triangle is 30 cm (Given).

The length of the sides which are equal is12 cm.

Let the third side length be 'a cm'.

Then, $Perimeter = a+b+c$

$\Rightarrow 30= a+12+12$

$\Rightarrow a = 6cm$

So, the semi-perimeter of the triangle is given by,

$s= \frac{1}{2}Perimeter =\frac{1}{2}\times30cm = 15cm$

Therefore, using Herons' Formula, calculating the area of the triangle

$A = \sqrt{s(s-a)(s-b)(s-c)}$

$= \sqrt{15(15-6)(15-12)(15-12)}$

$= \sqrt{15(9)(3)(3)}$

$= 9\sqrt{15}\ cm^2$

Hence, the area of the triangle is $9\sqrt{15}cm^2.$

Heron formula is used to find the area of a triangle by only using the sides of the triangle.

The s stands for the triangle's semiperimeter in Heron's formula and a,b,c represent the sides of the triangle whose area we have to find

half the perimeter of the triangle (s)=(a+b+c)/2

According to NCERT solutions for Class 9 Maths exercise 12.1, we can use the heron formula when the height of the triangle is not given. And heron formula is not useful for the right-angled triangle , equilateral triangle and isosceles triangle but the main application or use of herons formula is for scalene triangle.

In NCERT book exercise 12.1 Class 9 Maths we covered six questions and before exercise 12.1 Class 9 Maths we have solved three examples.

Also Read| Heron's Formula Class 9 Notes

## Benefits of NCERT Solutions for Class 9 Maths Exercise 12.1

• Class 9 Mathematics chapter 12 activity 12.1 offers the basics of the knowledge we'll need to go on to Class 9 Maths chapter 12 exercise 12.2.

• Heron's formula for calculating the perimeter or area of a triangle does not rely on the area formula that uses base and height.

• Heron's formula is also useful in trigonometry, as it can be used to calculate the law of cosines or the law of cotangents, among other things.

• In contrast to previous triangle area calculations, the heron formula does not need you to compute angles or other distances in the triangle first.

UNSAT 2023

## Key Features of Exercise 12.1 Class 9 Maths

1. Introduction to Heron's Formula: Class 9 maths ex 12.1 introduces students to Heron's Formula, a vital concept for finding the area of triangles when the lengths of their sides are known.

2. Step-by-Step Solutions: NCERT Solutions for class 9 ex 12.1 provide detailed, step-by-step explanations for each problem, helping students understand and apply Heron's Formula effectively.

3. Practical Application: The ex 12.1 class 9 offers problems that require students to calculate the areas of triangles, providing real-world applications of the formula.

4. Clear and Understandable Language: Class 9 ex 12.1 solutions are written in clear and understandable language to make the concepts accessible to students.

6. Preparation for Advanced Geometry: Heron's Formula is a fundamental concept that prepares students for more advanced topics in geometry they will encounter in higher classes.

## Subject Wise NCERT Exemplar Solutions

1. How many types of triangle are there used in NCERT solutions for Class 9 Maths exercise 12.1 ?

There are three types of triangle

• Right-angled triangle

• Isosceles triangle

• Scalene triangle

2. How do we find the area of a right-angled triangle according to NCERT solutions for Class 9 Maths exercise 12.1?

When the triangle is right-angled, we may use two sides containing the right angle as the base and height to apply the formula directly. these two sides are base and perpendicular

Area of a right-angled  triangle = 0.5 × perpendicular × base

3. How do we find the area of isosceles triangle according to NCERT solutions for Class 9 Maths exercise 12.1?

As two  sides of an isosceles triangle are equal, we find the midpoint of the non-equal side of the triangle. Then we connect that midpoint to the vertex above it, and the isosceles triangle is divided into two right-angle triangles. We can then get the area of each right-angle triangle and sum them to get the area of the isosceles triangle.

4. How do we find the area of a equilateral triangle triangle according to NCERT solutions for Class 9 Maths exercise 12.1?

As the equilateral triangle has all sides equal so it is easy to find the midpoint of the base side. Then we join that midpoint with the vertex above and this is how equilateral triangles get separated into two right-angle triangles then we can find the area of both right-angle triangles and then add them so as to get the area of an equilateral triangle

5. Give some real life example used throughout exercise 12.1 Class 9 Maths.

Making the traffic board

Area of a slide to get the quantity of paint required

6. What is the unit of measurement for length or breadth are taken in exercise 12.1 Class 9 Maths?

The metre (m) or centimetre (cm) are the units of measurement for length and width, respectively.

7. What is the unit of measurement for unit of measurement for area of any plane are taken in exercise 12.1 Class 9 Maths?

The area of any flat figure is measured in square metres (m^2), square centimetres (cm^2)

8. What kinds of questions do NCERT solutions for Class 9 Maths exercise 12.1 cover?

Exercise 12.1 Class 9 Maths covers all types of basic questions in which we have to apply the heron formula on triangles making real-life examples out of the question that make them interesting to solve but basic premise to apply formula only.

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##### Multimedia Specialist

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.

2 Jobs Available
##### Visual Communication Designer

Individuals who want to opt for a career as a Visual Communication Designer will work in the graphic design and arts industry. Every sector in the modern age is using visuals to connect with people, clients, or customers. This career involves art and technology and candidates who want to pursue their career as visual communication designer has a great scope of career opportunity.

2 Jobs Available
##### Copy Writer

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.

5 Jobs Available
##### Journalist

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available
##### Publisher

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available
##### Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. Ever since internet cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

3 Jobs Available
##### Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available

Advertising managers consult with the financial department to plan a marketing strategy schedule and cost estimates. We often see advertisements that attract us a lot, not every advertisement is just to promote a business but some of them provide a social message as well. There was an advertisement for a washing machine brand that implies a story that even a man can do household activities. And of course, how could we even forget those jingles which we often sing while working?

2 Jobs Available
##### Photographer

Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.

2 Jobs Available
##### Social Media Manager

A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.

2 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
##### Production Manager

Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

3 Jobs Available

A Team Leader is a professional responsible for guiding, monitoring and leading the entire group. He or she is responsible for motivating team members by providing a pleasant work environment to them and inspiring positive communication. A Team Leader contributes to the achievement of the organisation’s goals. He or she improves the confidence, product knowledge and communication skills of the team members and empowers them.

2 Jobs Available
##### Quality Systems Manager

A Quality Systems Manager is a professional responsible for developing strategies, processes, policies, standards and systems concerning the company as well as operations of its supply chain. It includes auditing to ensure compliance. It could also be carried out by a third party.

2 Jobs Available
##### Merchandiser

A career as a merchandiser requires one to promote specific products and services of one or different brands, to increase the in-house sales of the store. Merchandising job focuses on enticing the customers to enter the store and hence increasing their chances of buying a product. Although the buyer is the one who selects the lines, it all depends on the merchandiser on how much money a buyer will spend, how many lines will be purchased, and what will be the quantity of those lines. In a career as merchandiser, one is required to closely work with the display staff in order to decide in what way a product would be displayed so that sales can be maximised. In small brands or local retail stores, a merchandiser is responsible for both merchandising and buying.

2 Jobs Available
##### Procurement Manager

The procurement Manager is also known as  Purchasing Manager. The role of the Procurement Manager is to source products and services for a company. A Procurement Manager is involved in developing a purchasing strategy, including the company's budget and the supplies as well as the vendors who can provide goods and services to the company. His or her ultimate goal is to bring the right products or services at the right time with cost-effectiveness.

2 Jobs Available
##### Production Planner

Individuals who opt for a career as a production planner are professionals who are responsible for ensuring goods manufactured by the employing company are cost-effective and meets quality specifications including ensuring the availability of ready to distribute stock in a timely fashion manner.

2 Jobs Available
##### Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

3 Jobs Available
##### Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### ITSM Manager

ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

3 Jobs Available
##### .NET Developer

.NET Developer Job Description: A .NET Developer is a professional responsible for producing code using .NET languages. He or she is a software developer who uses the .NET technologies platform to create various applications. Dot NET Developer job comes with the responsibility of  creating, designing and developing applications using .NET languages such as VB and C#.

2 Jobs Available
##### Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available
##### DevOps Architect

A DevOps Architect is responsible for defining a systematic solution that fits the best across technical, operational and and management standards. He or she generates an organised solution by examining a large system environment and selects appropriate application frameworks in order to deal with the system’s difficulties.

2 Jobs Available
##### Cloud Solution Architect

Individuals who are interested in working as a Cloud Administration should have the necessary technical skills to handle various tasks related to computing. These include the design and implementation of cloud computing services, as well as the maintenance of their own. Aside from being able to program multiple programming languages, such as Ruby, Python, and Java, individuals also need a degree in computer science.

2 Jobs Available