Maths herons formula

THIS PPT IS ABOUT MATHS LESSON- HERON'S FORMULA FOR IX CLASS CBSE BOARD

1. HERON’S FORMULA

2. SELF INTRODUCTION NAME- Siddhi U Pawar CLASS- IX Arctic SUBJECT- Mathematics TOPIC- Heron’s Formula CONTENT- 1) Introduction 2) Derivation of the formula 3) Importance of the formula 4) Applications of the formula 5) Summary SCHOOL- Podar International School, Sangli.

3. INTRODUCTION We have studied in the earlier classes about the area of the triangles. We know that area of a triangle = ½ * base * height. This formula can be used only when we have the values of base and height given. But if length of three sides and one diagonal’s length is given of a triangle then we need another formula to find the area of the triangle. Therefore, hero of Alexandria created the formula known as heron’s formula – Where A is the area of the triangle, s= semiperimeter i.e. a + b + c / 2 and a, b, c and the sides of the triangle.

4. WHO DERIVED THIS FORMULA? Heron was born in about 10AD possibly in Alexandria in Egypt. His works on mathematical and physical are so numerous and varied that he was considered to be an encyclopaedic in writer in these field. His geometrical works deal with problems on mensuration written in three books. And in his Book I, he has derived the famous formula for the area of the triangle in terms of its three sides.

5. DERIVATION OF THE HERON’S FORMULA

6. FORMULA FOR EQUILATERAL TRIANGLE Let the side of the equilateral triangle be ‘a’. S = ½ (a + a + a) = 3/2 a A = √ s (s – a) (s – b) (s – c) = √ 3/2 a (3/2 a – a) (3/2 a – a) (3/2 a – a) = √ 3/2a (a/2) (a/2) (a/2) = √ 3 * a 2 * a 2 / 2 2 * 2 2 = a * a/4 √3 = a 2 /4 √3 = √3/4 a 2

7. IMPORTANCE OF THIS FORMULA There are just many figures that specifies the lengths and their area is to be found. Sometimes on the basis of perimeter just we need to find the area. All the types of triangles can find their area with this formula. Also sometimes quadrilaterals are bisected and their area is to be found. At that same time we can use this heron’s formula. We should first divide the quadrilateral and then we should find the area. All quadrilaterals’ area can be found by this formula.

8. APPLICATIONS OF THE FORMULA This formula is very useful so now let see its application Here cloth which is used, its area can be calculated by heron’s formula To calculate the small triangular shaped umbrella design etc.

9. APPLICATIONS OF THE FORMULA This formula is very helpful where it is not possible to find the height of a triangle. W e can find the area of small triangles in wheel toys We can find area of the quadrilaterals by dividing it into the triangles

10. PROBLEMS ON FORMULA There is a slide in a park. One of its side walls has been painted in some colour with a message ‘KEEP THE PARK GREEN AND CLEAN’ (See fig). If the sides of the wall ae 15m, 11m and 6m, find the area painted in colour. ANS. = a = 15m, b = 11m, c = 6m. A = √ s (s – a) (s – b) (s – c) = √16 (16 – 15) (16 – 11) (16 – 6) m 2 = √16 * 1 * 5 * 10 m 2 = √2 * 400 m 2 = 20 √2 m 2 Thus, the required area painted in colour = 20 √2 m 2

11. PROBLEMS ON FORMULA Q. Find the area of a quadrilatersl ABCD in which AB = 3cm, BC = 4cm, CD = 4cm, DA = 5cm and AC = 5cm. ANS.= Area of ABC tri. = 6 cm 2 Area of ACD tri. = 9.2 cm 2 Thus, area of ABCD = 6 + 9.2 = 15.2 cm 2

12. SUMMARY Area of a triangle with its side as a, b and c is calculated by using heron’s formula – where s = a + b + c / 2. Area of a quadrilateral whose sides and one diagonal are given, can be calculated by dividing the quadrilateral into two triangles and using the Heron’s formula.