Heron’s Formula

Heron’s formula It is particularly useful when it is not possible to find the height of the triangle easily.

The Formula:

Area = √s(s – a)(s – b)(s – c)

Where

s = (a + b + c)/2

Key Points

• Semi-perimeter: Half the perimeter of a triangle.
• Application: Can be used to find the area of any triangle, regardless of its shape.
• Derivation: The formula can be derived using the Pythagorean theorem and the properties of triangles.

In essence, Heron’s formula provides a direct method to calculate the area of a triangle when all side lengths are known, without requiring the calculation of height or other trigonometric functions.

Exercise 10.1

1. A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side a. Find the area of the signal board, using Heron’s formula.If its perimeter is 180 cm, what will be the area of the signal board?

Ans : 

Part 1: Using Heron’s Formula

• For an equilateral triangle,
•  all sides are equal, so a = b = c.
• Semi-perimeter (s) = (a + b + c)/2 = (a + a + a)/2 = 3a/2
• Using Heron’s formula: Area = √[s(s – a)(s – b)(s – c)] = √[(3a/2)(3a/2 – a)(3a/2 – a)(3a/2 – a)] = √[(3a/2)(a/2)(a/2)(a/2)] = (a²/4)√3

Therefore, the area of the equilateral triangle is (a²/4)√3 square units.

Part 2: Given Perimeter is 180 cm

• Perimeter of an equilateral triangle = 3a
• So, 3a = 180 cm
• a = 60 cm

Substituting a = 60 cm in the area formula:

• Area = (60²/4)√3 = 900√3 cm²

Therefore, the area of the traffic signal board is 900√3 cm².

2. The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m (see figure). The advertisements yield an earning of ₹5000 per m² per year. A company hired one of its walls for 3 months. How much rent did it pay?

Ans : 

Step 1: Calculate the area of the triangular wall 

Semi-perimeter (s) = (a + b + c)/2 = (122 + 22 + 120)/2 = 132 m

• Area = √[s(s – a)(s – b)(s – c)] = √[132(132-122)(132-22)(132-120)] = √[13210110*12] = 1210 m²

Step 2: Calculate the rent for the entire year

 Rent per m² per year = ₹5000 

Total area = 1210 m² 

Rent for the entire year = 5000 * 1210 = ₹6050000

Step 3: Calculate the rent for 3 months 

Rent for 3 months = (Rent for entire year) * (3/12) 

= 6050000 * (1/4) 

= ₹1512500

Therefore, the company paid ₹1512500 as rent.

3. There is a slide in a park. One of its side Company hired one of its walls for 3 months.walls has been painted in some colour with a message “KEEP THE PARK GREEN AND CLEAN” (see figure). If the sides of the wall are 15 m, 11 m and 6m, find the area painted in colour.

Ans : 

Since the wall is triangular, we can use Heron’s formula to find its area.

Heron’s formula:

• Area of a triangle = √(s(s-a)(s-b)(s-c)) Where:
  • s is the semi-perimeter (half the perimeter)

Calculations:

1. Calculate the semi-perimeter (s):
   • s = (a + b + c) / 2 = (15 + 11 + 6) / 2 = 32 / 2 = 16 m
2. Calculate the area using Heron’s formula:
   • Area = √(s(s-a)(s-b)(s-c)) 
   • = √(16 * (16-15) * (16-11) * (16-6)) 
   • = √(16 * 1 * 5 * 10)
   •  = √800 = 20√2 square meters

4. Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.

Ans : 

Step 1: Find the third side.

• Let the third side be ‘c’.
• Perimeter = a + b + c
  • 42 = 18 + 10 + c
  • c = 42 – 18 – 10 = 14 cm

Step 2: Use Heron’s formula 

• Area = √(s(s-a)(s-b)(s-c)) Where s is the semi-perimeter, calculated as s = (a + b + c) / 2
• Calculate the semi-perimeter:
  • s = (18 + 10 + 14) / 2 = 21 cm
• Calculate the area:
  • Area = √(21 * (21-18) * (21-10) * (21-14)) = √(21 * 3 * 11 * 7) = √(3 * 7 * 3 * 11 * 7) = 3 * 7 * √11 = 21√11 cm²

Therefore, the area of the triangle is 21√11 square centimeters.

5. Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm. Find its area.

Ans : 

Step 1: Find the sides

 Let the common ratio of the sides be ‘x’. So, the sides are 12x, 17x, and 25x cm.

Perimeter = 12x + 17x + 25x = 540 cm => 54x = 540 cm => x = 10 cm

Therefore, the sides of the triangle are:

• a = 12x = 12 * 10 = 120 cm
• b = 17x = 17 * 10 = 170 cm
• c = 25x = 25 * 10 = 250 cm

Step 2: Calculate the semi-perimeter

 Semi-perimeter (s) = (a + b + c) / 2 = (120 + 170 + 250) / 2 = 540 / 2 = 270 cm

Step 3: Use Heron’s formula

Area = √(s(s-a)(s-b)(s-c)) =

 √(270 * (270-120) * (270-170) * (270-250)) 

= √(270 * 150 * 100 * 20)

 = √(3 * 3 * 3 * 10 * 5 * 2 * 5 * 5 * 2 * 2 * 5 * 2)

 = 3 * 3 * 5 * 5 * 2 * 2 * √10 

= 900 √10 cm²

Therefore, the area of the triangle is 900√10 square centimeters.

6. An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

Ans : 

Step 1: Find the length of the third side

• Let the third side be ‘c’.
• Perimeter = a + b + c
  • 30 = 12 + 12 + c
  • c = 30 – 24 = 6 cm

Step 2: Use Heron’s 

• Area = √(s(s-a)(s-b)(s-c)) Where s is the semi-perimeter, calculated as s = (a + b + c) / 2

• Calculate the semi-perimeter:
  • s = (12 + 12 + 6) / 2 = 15 cm
• Calculate the area:
  • Area = √(15 * (15-12) * (15-12) * (15-6)) 
  • = √(15 * 3 * 3 * 9) = √1215 = 9√15 cm²

Therefore, the area of the isosceles triangle is 9√15 square centimeters.