HERONS FORMULA

HERONS FORMULAPRESENTED BY :Krishan KumarK. V. VikaspuriHow will you find the area of this triangle?.

24 cm 8 cm

This is a right angled triangle and we know that area of this triangle = *base *height

Area of this triangle = *8*24 =96 sq.cm How will you find the area of this equilateral triangle ?

10 cm 10 cm

10 cm

How will you find the area of this isosceles triangle ?

12 cm 12 cm

10 cm

We know the lengths of the sides of the scalene triangle and not the height. Can you still find its area ?Herons formulaHeron gave the famous formula for finding the area of a triangle in terms of its three sides.

The formula given by him is known as Herons formula.It is stated as : area of triangle = s(s-a)(s-b)(s-c)Where a,b and c are sides of the triangle ,and s= semi-perimeter i.e. S= a+b+c/2About HeronsHeron was born in about 10AD possibly in Alexandria in Egypt. His works on mathematical and physical subjects are so numerous and varied that he is considered to be an encyclopedic writer in this fields. His geometrical works deal largely with problems on mensuration written in three books. Book 1 deals with the area of squares, rectangles ,triangles, various other specialised quadrilaterals, the regular polygons . n this book ,Heron has derived famous formula for the area of a triangle in terms of its three sides.

Herons formula is helpful where we know the three sides of triangle and it is not possible to find the height of the triangle easily.To find the area of triangle using Herons formulaQ. Find the area of the triangle ,whose sides are 40 m, 24 m, 32 m .

Solution : let us take a =40 m, b = 24 m , c = 3 so that we have s = 40+24+32/2 = 48 ms a = (48 40) m = 8 m,s b = (48 24) m = 24 m,s c = (48 32) m = 16 m.Therefore, area of the triangle = s(s a) (s b) (s c)

= 48 8 24 16m2 = 384m2PROOF OF THE HERONS FORMULALet a, b and c be the lengths and h be the height to the side of length c

We have s = a+b+c 2 a h b 2s = a +b +c 2(s-a) =b+C-a p q 2(s-b) =a+c-b c 2(s-c) = a+b-c There is at least one side of our triangle for which the altitude lies "inside" the triangle. For convenience make that the side of length c.

p + q = c

Application of Herons Formula in Finding Areas of Quadrilaterals Area of a quadrilateral whose sides and one diagonal are given, can be calculated by dividing the quadrilateral into two triangles and using the Herons formula.

Sanya has a piece of land which is in the shape of a rhombus(see Fig. 12.13). She wants her one daughter and one son to work on the land andproduce different crops. She divided the land in two equal parts. If the perimeter ofthe land is 400 m and one of the diagonals is 160 m, how much area each of them will 100get for their crops? A DSolution : Let ABCD be the field.Perimeter = 400 mSo, each side = 400 m 4 = 100 m. 1601 100 mi.e. AB = AD = 100 m.Let diagonal BD = 160 m. B C

Then semi-perimeter s of ABD is given by 100 ms =100+ 100 +160/2 = 180 mTherefore, area of ABD = 180(180 100) (180 100) (180 160)= 180 80 80 20 m2 = 4800 m2Therefore, each of them will get an area of 4800 m2.

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