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Heron's Formula Notes: Class 9 Maths Chapter 12
Class 9 Maths Chapter 12 Heron's Formula covers finding the area of a triangle using its side lengths, and applying it to find areas of quadrilaterals and other polygons split into triangles.
Overview of Topics & Subtopics (Click to Jump)
1. Area of a Triangle by Heron's Formula
Short Note:
• For a triangle with sides a, b, c, first find the semi-perimeter: s = (a + b + c)/2.
• Heron's Formula: Area = √[s(s−a)(s−b)(s−c)].
• This formula is especially useful when the height of the triangle is not directly known, such as for scalene triangles.
2. Application to Other Shapes
Short Note:
• The area of a quadrilateral or other polygon can be found by splitting it into triangles using its diagonals, then applying Heron's formula to each triangle and adding the areas.
NCERT Textbook Exercise Questions
A triangle has sides 150 cm, 120 cm and 200 cm. Find its area.
Answer: s = (150 + 120 + 200)/2 = 235 cm. Area = √[235(235−150)(235−120)(235−200)] = √[235 × 85 × 115 × 35] ≈ 8966.56 cm² (approx.).
Find the area of a triangle whose sides are 18 cm, 10 cm and 24 cm.
Answer: s = (18+10+24)/2 = 26. Area = √[26(26−18)(26−10)(26−24)] = √[26 × 8 × 16 × 2] = √6656 ≈ 81.6 cm².
The sides of a triangular plot are in the ratio 3:5:7 and its perimeter is 300 m. Find its area.
Answer: Sides = 3x, 5x, 7x with 15x = 300 ⇒ x = 20, so sides = 60, 100, 140 m. s = 150. Area = √[150(150−60)(150−100)(150−140)] = √[150 × 90 × 50 × 10] = √6750000 = 1500√3 m².
A rhombus-shaped field has green grass for 18 cows to graze; each side of the rhombus is 30 m and the diagonal is 48 m. Find the area available for each cow.
Answer: Split the rhombus into two congruent triangles by the 48 m diagonal, each with sides 30, 30, 48. s = 54. Area of one triangle = √[54 × 24 × 24 × 6] = 432 m². Total rhombus area = 864 m². Area per cow = 864/18 = 48 m².
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