Heron's Formula
Introduction to Heron's Formula
→Heron’s formula is used to calculate the area of any triangle with the length of all three sides given.
where:
- s is the semi-perimeter of the triangle, calculated as (a + b + c)/2
- a, b, and c are the lengths of the sides of the triangle
Exercise- 10.1
Question 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?
Question 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. 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?
Rent = 1320 × 1250 = ₹1650000
Therefore, the company paid ₹16,50,000 for 3 months.
Question 3:
There is a slide in a park. One of its side walls has been painted with a message “KEEP THE PARK GREEN AND CLEAN”. If the sides of the wall are 15 m, 11 m, and 6 m, find the area painted in colour.
Question 4:
Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
Solution:
Given:
c = P - (a + b)
c = 42 - (18 + 10) = 14 cm
Now, apply Heron's Formula:
Semi-perimeter s is given by:
s = P / 2 = 42 / 2 = 21 cm
Heron's Formula for area A:
A = √[s(s - a)(s - b)(s - c)]
A = √[21(21 - 18)(21 - 10)(21 - 14)]
A = √[21 × 3 × 11 × 7]
A = √[4851] ≈ 69.7 cm²
So, the area of the triangle is approximately 69.7 cm²
- Side a = 18 cm
- Side b = 10 cm
- Perimeter P = 42 cm
c = P - (a + b)
c = 42 - (18 + 10) = 14 cm
Now, apply Heron's Formula:
Semi-perimeter s is given by:
s = P / 2 = 42 / 2 = 21 cm
Heron's Formula for area A:
A = √[s(s - a)(s - b)(s - c)]
A = √[21(21 - 18)(21 - 10)(21 - 14)]
A = √[21 × 3 × 11 × 7]
A = √[4851] ≈ 69.7 cm²
So, the area of the triangle is approximately 69.7 cm²
Question 5:
Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm. Find its area.
Question 6:
An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.
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