 Hello and welcome to the session. In this session we will discuss hero's formula. First we shall discuss how we find the area of a triangle by hero's formula. We know that for any triangle ABC its area is given by half into base into height. So for this it would be half into base which is BC into the height that is the altitude AM. So using this formula the area of the triangle can be easily found out if we are given the height and the base of the triangle. But if we are given the measures of the three sides of a triangle then we will use the hero's formula to calculate its area. So area of a triangle is equal to square root S multiplied by S minus A multiplied by S minus B multiplied by S minus C where we have these ABC are the sides of the triangle where S is the semi-perimeter that is it is half the perimeter of the triangle. So we have S is equal to A plus B plus C upon 2. This formula can be used where it is not possible for us to find the height of the given triangle easily. Suppose that we need to find the area of a triangle whose sides are of length. Let's take A equal to 52 centimeters, B equal to 56 centimeters and C equal to 60 centimeters. That is these are the lengths of three sides of a triangle. First let's find out the semi-perimeter that is S. This is equal to A plus B plus C upon 2 that is 52 plus 56 plus 60 upon 2 which is equal to 168 upon 2 that is 84 centimeters. So we have got the semi-perimeter S equal to 84 centimeters. Now we shall find out S minus A this is equal to 84 minus 52 that is equal to 32 centimeters. Then S minus B is equal to 84 minus 56 and this is equal to 28 centimeters. Then S minus C equal to 84 minus 60 and that is equal to 24 centimeters. Now we have area of triangle is equal to square root S that is 84 multiplied by S minus A that is 32 multiplied by S minus B that is 28 multiplied by S minus E that is 24. And this comes out to be equal to 1, 3, 4, 4 centimeters square. So this is how we find the area of a triangle by Hero's formula. Next we see how we find the area of a quadrilateral by Hero's formula. Now the area of a quadrilateral whose sides and one diagonal are given can be calculated by dividing the quadrilateral into two triangles and using the Hero's formula. So to find the area for the quadrilateral ABCD what we do is we have divided this quadrilateral into two triangles triangle ABD and triangle BCD. So we will find the area of both these triangles by Hero's formula and add them to get the area of the given quadrilateral. Like we need to find the area of the quadrilateral ABCD in which we are given AB is 3 centimeters, BC is 4 centimeters, CD is 4 centimeters and DA is 5 centimeters and the diagonal AC of the quadrilateral is given as 5 centimeters. So here we will consider the triangles ABC and triangle ACD. First let's consider triangle ABC. In this let's take the side A equal to 5 centimeters, B equal to 4 centimeters and C equal to 3 centimeters. So for this triangle ABC, let's find out the semi-perimeter S equal to A plus B plus C upon 2. This is equal to 6 centimeters. Now then by Hero's formula we have area of triangle ABC is equal to square root S that is 6 multiplied by S minus A multiplied by S minus B multiplied by S minus C that is equal to square root 6 into 1 into 2 into 3 and this is equal to 6 centimeters square. Now we consider the triangle ACD in this we take A dash equal to 5 centimeters, side B dash equal to 5 centimeters and C dash equal to 4 centimeters. So for the triangle ACD we take the semi-perimeter S dash equal to A dash plus B dash plus C dash upon 2 and this is equal to 7 centimeters. Then by Hero's formula we have area of triangle ACD is equal to square root S dash that is 7 multiplied by S dash minus A dash multiplied by S dash minus B dash again multiplied by S dash minus C dash. This is further we equal to square root 7 multiplied by 2 multiplied by 2 multiplied by 3 and this is equal to 2 root 21 centimeters square. This is the area of triangle ACD and this is equal to 9.2 centimeters square approximately. Now the area of quadrilateral ABCD is equal to the area of triangle ABC plus area of triangle ACD and this is equal to 6 that is the area of triangle ABC plus 9.2 which is the area of triangle ACD and that is equal to 15.2 centimeters square approximately. So this is how we find the area of a quadrilateral by dividing it into two triangles and using the hero's formula. This completes this session. Hope you have understood how we find the area of a triangle and area of quadrilateral by hero's formula.