 Hello my dear friends, so I'm welcoming again all of you to this new session on a topic called Heron's Formula. So we are going to discuss how to find out the area of a triangle if three side lengths of the triangle are given. Now before we look into all that, let us understand whatever we have studied so far in terms of finding area in a triangle. So all of you would be familiar that we have already learned a formula for finding out the area of a triangle and that is given by area of triangle is nothing but half into base into height. Now this formula you have been using for a while now, what does it mean? It means that if there is a triangle given, then if you drop a perpendicular which is also called an altitude from any of the vertices. So let's say this triangle is A, B, C and this side is H, let's say AD. AD is perpendicular to BC and AD is equal to H and BC length is B, BC is equal to B, then my friend area of triangle ABC, triangle ABC, so we just denoted by like this AR within brackets triangle ABC is equal to half into B into H. Is it B and H could be different for different sides. For example, if this was HAD, so you can also drop a perpendicular like that and let's say this is H1 and let's say this side is B1 then this also will be equal to half into B1 into H1 and similarly if you drop a perpendicular from C on to AB let's say this is again H2 and this is B2 then this will also be equal to half into B2 into H2. So base and corresponding altitude if you multiply and divide by 2 you will get the area of triangle. But what if we do not have the length of the altitude or instead of that we have all the three lengths then there was a guy called Heron or actually the name is Hero and this guy was a Greek mathematician. Now he was from a place called Alexandria which is in current day Egypt in Africa. So this guy was Hero of Alexandria the name goes like this Hero of Alexandria and he only figured out this method of finding out area of a triangle which we are going to discuss now. Now interestingly he was not the first person who actually did it but before that also you know people knew about it and today we know that Hero's formula or Heron's formula as you call it is a special case of special case of something called Brahma Gupta's formula. So Brahma Gupta's formula was nothing but a method or a formula for finding out the area of a cyclic quadrilateral. What is a cyclic quadrilateral? So if you take any circle and pick four points on it one two three four and join them you'll get a quadrilateral. This quadrilateral is called a cyclic quadrilateral. So cyclic quadrilateral is nothing but quadrilateral inscribed within a circle that's called cyclic quadrilateral. So hence if let's say A, B, C, D is the cyclic quadrilateral and let's say the sides are A, B, C and D side lengths are A, B and C and D then he found out area of quadrilateral A, B, C, D is equal to under root S minus A S minus B times S minus C times S minus B. Now you'll ask what is S? Now S is called semi-perimeter semi-perimeter of the quadrilateral right? So hence you can say semi means half so half of perimeter that means A plus B plus C plus D. So if S is this then area of quadrilateral is given by this formula. Now interestingly this formula also is a specific case of another formula. Now this particular formula holds only for cyclic quadrilateral. So if you have to find out the area of a cyclic quadrilateral then this formula will hold. But this is also again a generic case of something called Brett Snyder's formula. So hence Brahma Gupta formula is also a specific case of something called Brett Snyder's formula. So those who know trigonometry or they will be able to understand this or comprehend this. Hence what is the formula then? So if you see if you and what is this formula about? Now this formula talks about any quadrilateral in general. So you don't need to you know have cyclic quadrilateral. So what I'm saying is Brett Snyder's formula is valid for any quadrilateral. So let's say this quadrilateral is A, B, C, and D. And you join or rather let this angle be alpha, let this be beta, let this angle be gamma and let this be delta. And again sides are A, B, C, and D. Then area of triangle is not sorry not triangle area of quadrilateral. A, B, C, D is given by is given by the formula this S minus A, S minus B, S minus C, S minus D and within under root only minus A times B times C times D and then cos square alpha plus gamma by 2. Okay, this is what the formula is. Let me rewrite it afresh so that you don't get confused. So the area is nothing but again I'm writing like this S minus A, S minus B, S minus C, S minus D like a previous case, but there is an additional term here which is A, B, C, D and then cos square sum of or half the sum of the opposite angle. So hence alpha plus gamma by 2. You could have taken beta and delta also the result will be same why because alpha plus beta plus gamma plus delta is 360 degrees and if you know about trigonometry then you know cos of 180 minus theta is cos theta minus cos theta and you square it you can again get the same thing. So let's not get into the details of this formula. This is for just for information sake that if you know opposite angles of a quadrilateral and if you know the sides of a quadrilateral then you will be able to find out the area of the quadrilateral. Now what we are talking about so hence this is just for your information and we now we are saying that Brett Snyder's formula was a general case of something called Brahma Gupta's formula and Brahma Gupta's formulas let's say one specific case is Heron's formula but what is this Heron's formula by the way. Okay so let me write it here. So now if this is a triangle then Heron's formula say area of triangle ABC whose sides a b and c small a b and c are given is equal to under root s s minus a s minus b s minus c isn't it where s is again semi-perimeter which is nothing but a plus b plus c upon 2 right. So this is what is Heron's formula for a triangle. Now you can ask how is this specific case from a Brahma Gupta's formula. So if you see in a triangle there is no fourth side so hence in that case d will be zero correct for a triangle d is equal to zero or not the fourth side is nothing it's zero so hence if you put d is equal to zero in this formula you will get nothing but this formula doesn't it you can try for a cyclic quadrilateral what was it cyclic quadrilateral was nothing but under root s minus a s minus b s minus c and s minus d all under root and s was in that case a plus b plus c plus d by 2 so if you put d is equal to zero s will become a plus b plus c upon 2 and now if you put d equal to zero here as well it will be reduced only to s and you will eventually get this formula isn't it. So this is Heron's formula in the next session let us understand how do we derive Heron's formula and in the later subsequent sessions we'll take up applications of Heron's formula. Thank you.