What is Heron\

Let us learn about Heron's Formula and its application. Area of different types of triangles, such as scalene, isosceles, and equilateral triangles, can be found using Heron's formula.

We need base and height every time we want to calculate the area of a triangle. But if we have a measure of sides instead of height and we have to find the area then what?

Now suppose that we know the lengths of the sides of a scalene triangle and not the height. Can we still find its area?

Yes, the best way to find the area in such a situation is to use heron's formula. The formula given by Heron about the area of a triangle is also known as Heron's formula.

What is Heron's formula?

If a, b and c are the three sides of a triangle, respectively, then Heron's formula is given by:

Area Of a Triangle: ?s(s-a)(s-b)(s-c), where a, b and c are the sides of the triangle and s=(a+b+c)/2.

Now we have 2 formulas to calculate the area of triangles. With this formula, we can find the area of triangles where it is not possible to find the height of the triangle easily. Let us apply it to find the area of triangles with given sides.

Heron's Formula Examples

Example: Find the area of a triangle, two sides of which are 8 cm and 11 cm and the perimeter is 32 cm.

Solution : perimeter of the triangle = 32 cm, a = 8 cm and b = 11 cm.

Third side c = 32 cm � (8 + 11) cm = 13 cm

So, 2s = 32 i.e. s = 16 cm,

s � a = (16 � 8) cm = 8 cm,

s � b = (16 � 11) cm = 5 cm,

s � c = (16 � 13) cm = 3 cm.

Therefore, area of the triangle = ?s (s-a) (s-b) (s-c)

= ?16 x 8 x 5 x3

= 8?30 cm²

Example: A triangular park ABC has sides 120m, 80m and 50m. A gardener Dhania has to put a fence all around it and also plant grass inside. How much area does she need to plant? Find the cost of fencing it with barbed wire at the rate of Rs 20 per metre leaving a space 3m wide for a gate on one side.

Heron's Formula

Solution : We have

2s = 50 m + 80 m + 120 m = 250 m.

i.e., s = 125 m

Now, s � a = (125 � 120) m = 5m,

s � b = (125 � 80) m = 45 m,

s � c = (125 � 50) m = 75 m.

Therefore, area of the triangle = ?s (s-a) (s-b) (s-c)

= ?125�5 �45 �75 m²

= 375?15 m²

Also, perimeter of the park = AB + BC + CA = 250 m

Therefore, length of the wire needed for fencing = 250 m � 3 m (to be left for gate) = 247 m And so the cost of fencing = Rs 20 � 247 = Rs 4940

Example: The sides of a triangular plot are in the ratio of 3 : 5 : 7 and its perimeter is 300 m. Find its area.

Heron's formula

Solution : Suppose that the sides, in metres, are 3x, 5x and 7x

So 3x + 5x + 7x = 300 (perimeter of the triangle)

Therefore, 15x = 300,

x = 20

So the sides of the triangle are 3 � 20 m, 5 � 20 m and 7 � 20 m i.e., 60 m, 100 m and 140 m.

s=(60+100+140)/2m=150m.

Therefore, area =?s (s-a) (s-b) (s-c)

=?150(150-60)(150-100)(150-140) m²

=?150 X 90 X 50 X 10 m²

=1500?3m²

Application of Heron's Formula

We will see the application of heron's formula in finding the area of the quadrilateral. Many a times it is difficult to find the area of a quadrilateral directly. So we need to divide the quadrilateral in triangular parts and then use the formula for area of the triangle.

Example: Kamla has a triangular field with sides 240 m, 200 m, 360 m, where she grew wheat. In another triangular field with sides 240 m, 320 m, 400 m adjacent to the previous field, she wanted to grow potatoes and onion.

She divided the field in two parts by joining the mid-point of the longest side to the opposite vertex and grew potatoes in one part and onions in the other part. How much area (in hectares) has been used for wheat, potatoes and onions?

Heron's Formula

Solution: Let ABC be the field where wheat is grown. Also let ACD be the field which has been divided into two parts by joining C to the mid-point E of AD. For the

area of triangle ABC, we have

a = 200 m, b = 240 m, c = 360 m

s= (200+240+360)/2 = 400m

So, the area for growing wheat

= ?400(400-200) (400 � 240) (400 � 360) m²

=?400 x 200 x 160 x 40m² =16000?2m² =1.6?2 hectares

Area of ?ACD��

we have s = (240+320+400)/2 m =480m

So area of ?ACD = ?480(480 - 240) (480 -320) (480 - 400)
= ?480 x 240 x 160 x 80 = 38400m² =3.84 hectare.

Therefore, area for growing potatoes = area for growing onions = (3.84 � �2) hectares = 1.92 hectares.

Example: Sanya has a piece of land that is in the shape of a rhombus. She wants her one daughter and one son to work on the land and produce different crops. She divided the land in two equal parts. If the perimeter of the land is 400 m and one of the diagonals is 160m, how much area each of them will get for their crops?

Heron's Formula