Heron's Formula for Class 9

Heron of Alexandria was a great mathematician who was the first to derive the formula of calculation of the area of a triangle using the length of all three sides. This formula is now known as Heron’s Formula or Heron’s Formula. Heron of Alexandria also used this formula to derive the area of quadrilaterals as well as high-order polygons. This formula is widely used in trigonometry as well as proving the law of cosines, the law of cotangents, etc. However, angle measurement is not required to calculate its area.

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What is Heron’s Formula?

Heron’s formula is used to determine the area of a triangle using the lengths of all its three sides. This formula is also used to determine the area of quadrilaterals as well as high-order polygons.

Heron’s Formula for Finding Area of a Triangle

According to Heron and his formula, we can determine the area of any triangle be it scalene, equilateral, or isosceles with the help of the formula as long as we have the lengths of all three sides provided.

For example, if we have a triangle XYZ and the sides are x, y, and z respectively, then the area of the triangle will be as follows- 

Area=√s(s−x)(s−y)(s−z)

Area=√s(s−x)(s−y)(s−z)

Here s refers to the semi-perimeter of the triangle i.e. s = (x + y + z)/2.

Heron’s Formula for Equilateral Triangle

To derive the area of an equilateral triangle, first, we have to find out the semi-perimeter of the triangle i.e.

 s = (x + x + x)/2

s = 3x/2

Here x is the length of the sides of the triangle.

By putting it in Heron’s formula, we get

A = √[s(s-x)3]

Heron’s Formula for Isosceles Triangle

An isosceles triangle is a triangle whose two sides are equal while those corresponding angles are congruent to each other. We can find out the area of an isosceles triangle from Heron’s Formula in the following way -

Let the length of the congruent sides be x and the length of the base be y.

So, their semi-perimeter s = (x + x + y)/2

s = (2x + y)/2

Area of a triangle according to Heron’s Formula

Area = √s(s−x)(s−y)(s−z)

 s(s−x)(s−y)(s−z)

In an isosceles triangle,

Area = √s(s−x)(s−y)(s−x)

s(s−x)(s−y)(s−x)

 = √[s(s - x)2(s - y)]

 = (s - x)√s(s−y)s(s−y)

Area of a Triangle Using Heron’s Formula

There are two steps involved in finding out the area of a triangle. The steps are as follows -

• First, we have to find out the value of the semi-perimeter of the triangle given to us.

• Next, we have to use Heron's formula, substitute the values obtained and given, and thus find out the area of the triangle.

Solved Example

A triangle XYZ has three sides of length 4cm, 13cm, and 15 cm. Calculate the area of the triangle XYZ using Heron’s Formula.

Solution: Semiperimeter of the triangle XYZ s = (4 + 13 + 15)/2 = 32/2 = 16

By using Heron’s Formula, we get,

Area = √s(s−x)(s−y)(s−z)

s(s−x)(s−y)(s−z)

= √16(16−4)(16−13)(16−15)

16(16−4)(16−13)(16−15)

= √(16 * 12 * 3 * 1)

= √576

 = 24 sp. Cm.

Heron's Formula for Class 9

From the above mentioned text and the examples, you must have understood the Heron's Formula.

If you want to try your hands on more solved and unsolved questions, you can visit the website of Vedantu or the mobile app which is easily available on both play store and app store.

Since Mathematics is more of a practical subject, some students find it very interesting and some find it very difficult. For the ones who find it to be tough, there's a little reminder for you, that with practice, nothing is unattainable. 

Especially, when you talk about such subjects, only practice can help you ace them. 

Let us look at some of the easy tricks that can be used by the students to prepare for a polarising subject like Maths, which is loved by some and loathed by others. 

• Practice Makes You Better 

Just like any other task in life, you get better at things when you practice, whether it's about learning to swim or to dance or to drive, anything, you will ace it with time. Nobody is born with a talent, you get to master a particular skill when you repeatedly do that. And, this law works for the subject of Mathematics too. Hence, students are advised to be in regular touch with the subject. They shall be practising the questions regularly so that they don't lose focus and forget the basic concepts. 

• Make It a Habit to Review Your Mistakes 

Reviewing your mistakes is as important as focusing on understanding the concepts. When you review your mistake, you go to its root and check where you went wrong. This allows you to think about it again and understand things better. Once you get to know about it, your mind registers it and from next time, you don't do that again. To avoid making the mistakes over and over again, you may try using the step by step approach which will surely get you the right answer. 

• Focus on the Concepts, And Not the Processes 

Focusing on the concept is more important than focusing on the process. Understanding of the concept should be your first priority, you may not just cram the process and don't care about the resources used. Once you get to know the concept, you understand the process on your own. To understand the logic behind each solution is a must as Maths is a kind of a sequential subject which might need the logic behind to solve the next one. 

• Use Real-World Examples 

Examples can make your preparation process easy and quick. Your mind remembers the examples of mnemonics for a longer period of time. And that too if you use real world examples which are around you, the process would become faster and easier. The best solution to any maths problem is to apply it to your daily life situation and then look into it. Students shall try doing it at all times and the results are surely going to surprise them! 

• Take Help From a Study Buddy 

Taking help from your friends makes it simpler to understand the concepts. You should work on your and partners weaknesses together and solve each other's doubts. This makes you feel relaxed and prepared. You should make a study buddy and motivate each other. If one is lagging behind, then you may help him pull up and vice versa. This is one of the most effective ways to master the subject. 

Hope you might have liked these tricks, if you will be applying this while preparing for Maths, you will always find them useful. Keeping these points in mind will surely give you great results. Maths is a subject which has a lot of importance, not just for exams, but for life, so, whether you like it or hate it, you have to do it at some point of time and at that moment you must use these techniques which will help you learn the things faster.