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# HeronŌĆÖs Formula

How will you find the area of a triangle, if the height is not known but the lengths of the three sides are known?

HeronŌĆÖs Formula

How will you find the area of a triangle, if the height is not known but the lengths of the three sides are known?

For this, Heron has given a formula to find the area of a triangle.

If a, b and c are the sides of a triangle, then the area of a triangle = ŌłÜ[ s ( s ŌłÆ a )(s ŌłÆ b)(s ŌłÆ c)] sq.units.

where s = [ a + b + c ] / 2, ŌĆśsŌĆÖ is the semi-perimeter (that is half of the perimeter) of the triangle. Note

If we assume that the sides are of equal length that is a = b = c, then HeronŌĆÖs formula will be ŌłÜ3/4 a2 sq.units, which is the area of an equilateral triangle.

Example 7.1

The lengths of sides of a triangular field are 28 m, 15 m and 41 m. Calculate the area of the field. Find the cost of levelling the field at the rate of Ōé╣ 20 per m2.

Solution

Let a = 28 m, b = 15 m and c = 41m

Then, s = (a + b + c) / 2 = (28 +15+ 41) / 2 = 84/2 = 42 m

Area of triangular field = ŌłÜ [ s(s ŌłÆ a)(s ŌłÆ b)(s ŌłÆ c) ]

= ŌłÜ [ 42(42 ŌłÆ 28)(42 ŌłÆ15)(42 ŌłÆ 41)]

= ŌłÜ [42 ├Ś14 ├Ś 27 ├Ś1]

= ŌłÜ [ 2├Ś3├Ś7├Ś7├Ś2├Ś3├Ś3├Ś3├Ś1]

= 2├Ś3├Ś7├Ś3

= 126 m2

Given the cost of levelling is Ōé╣ 20 per m2.

The total cost of levelling the field = 20 ├Ś126 = Ōé╣ 2520.

Example 7.2

Three different triangular plots are available for sale in a locality. Each plot has a perimeter of 120 m. The side lengths are also given: Help the buyer to decide which among these will be more spacious.

Solution

For clarity, let us draw a rough figure indicating the measurements: (i) The semi-perimeter of Note that all the semi-perimeters are equal.

(ii) Area of triangle using HeronŌĆÖs formula:

In Fig.7.4, Area of triangle = ŌłÜ [60(60 ŌłÆ 30)(60 ŌłÆ 40)(60 ŌłÆ 50)]

= ŌłÜ [60├Ś30├Ś20├Ś10]

= ŌłÜ [30├Ś2├Ś30├Ś2├Ś10├Ś10]

= 600 m2

In Fig.7.5, Area of triangle = ŌłÜ [60(60 ŌłÆ 35)(60 ŌłÆ 40)(60 ŌłÆ 45)]

= ŌłÜ [60├Ś25├Ś20├Ś15]

= ŌłÜ [20├Ś3├Ś5├Ś5├Ś20├Ś3├Ś5]

= 300 ŌłÜ5 ( Since ŌłÜ5 = 2.236 )

= 670.8 m2

In Fig.7.6, Area of triangle = ŌłÜ [60(60 ŌłÆ 40)(60 ŌłÆ 40)(60 ŌłÆ 40)]

= ŌłÜ [60├Ś20├Ś20├Ś20]

= ŌłÜ [3├Ś20├Ś20├Ś20├Ś20]

= 400 ŌłÜ3 ( Since ŌłÜ3 = 1.732 )

= 692.8 m2

We find that though the perimeters are same, the areas of the three triangular plots are different. The area of the triangle in Fig 7.6 is the greatest among these; the buyer can be suggested to choose this since it is more spacious.

Note

If the perimeter of different types of triangles have the same value, among all the types of triangles, the equilateral triangle possess the greatest area. We will learn more about maximum areas in higher classes.

Tags : Mensuration | Maths , 9th Maths : UNIT 7 : Mensuration
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9th Maths : UNIT 7 : Mensuration : HeronŌĆÖs Formula | Mensuration | Maths