Important Formulas, Rules & Key Points

• Semi-perimeter (s): s = (a + b + c) / 2
• Heron's Formula for Area of Triangle: Area = √(s(s - a)(s - b)(s - c))
• Equilateral Triangle Area: (√3 / 4) × side²

Question 1:

Find the area of a triangle whose sides are 13 cm, 14 cm, and 15 cm.

Solution:
Let the sides be a = 13 cm, b = 14 cm, c = 15 cm.
First, calculate semi-perimeter, s = (13 + 14 + 15) / 2 = 42 / 2 = 21 cm.
Applying Heron's formula:
Area = √(21 × (21 - 13) × (21 - 14) × (21 - 15))
= √(21 × 8 × 7 × 6)
= √(3 × 7 × 2 × 4 × 7 × 2 × 3)
= √(3² × 7² × 2² × 2²)
= 3 × 7 × 2 × 2 = 84 cm².
Answer: 84 cm²

Question 2:

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

Solution:
Let the sides be 3x, 5x, and 7x.
Perimeter = 3x + 5x + 7x = 300 m
15x = 300 ⇒ x = 20 m.
Sides are: a = 60 m, b = 100 m, c = 140 m.
Semi-perimeter, s = 300 / 2 = 150 m.
Area = √(150 × (150 - 60) × (150 - 100) × (150 - 140))
= √(150 × 90 × 50 × 10)
= √(15000000) = 1500√3 m².
Answer: 1500√3 m²

Aptitude Corner

Quick Trick: When solving Heron's formula problems, avoid multiplying large numbers inside the square root. Instead, break every number down into prime factors to easily group identical factors in pairs.

🔍 Search more MP Board Solutions & Exercises