Class 9 Maths Chapter 12 Heron's Formula NCERT Solutions: All Exercises Step-by-Step Area Calculations

Introduction

Chapter 12 gives you Heron's formula to find triangle area when you know all three sides (no height needed). Perfect when base/height hard to measure. Also covers equilateral triangles and applications like paths around fields.

Two exercises with practical problems. Easy to memorize, quick to calculate. Key for board exams.

Key Formula

Heron's Formula:

Area = sqrt[s(s-a)(s-b)(s-c)]
where s = (a+b+c)/2 (semi-perimeter), a,b,c = sides

Equilateral triangle: Area = (sqrt(3)/4) × side²

Exercise 12.1 Solved Questions

Question 1: Sides 13cm, 14cm, 15cm. Find area.

Solution:
s = (13+14+15)/2 = 21 cm
Area = sqrt[21(21-13)(21-14)(21-15)] = sqrt[21×8×7×6]
= sqrt = 84 cm²

Question 2: Sides 5cm, 12cm, 13cm. Area?

Solution: Right triangle! s = (5+12+13)/2 = 15
Area = sqrt[15(15-5)(15-12)(15-13)] = sqrt[15×10×3×2] = sqrt = 30 cm²

Question 3: Equilateral triangle side 6cm.

Solution: Area = (sqrt(3)/4) × 6² = (sqrt(3)/4) × 36 = 9sqrt(3) cm² ≈ 15.58 cm²

Question 4: Sides 7cm, 8cm, 10cm.

Solution: s = (7+8+10)/2 = 12.5
Area = sqrt[12.5(12.5-7)(12.5-8)(12.5-10)] = sqrt[12.5×5.5×4.5×2.5]
= sqrt[775.78125] ≈ 27.86 cm²

Question 5: Path 25m, 30m, 40m around triangle field. Area?

Solution: s = (25+30+40)/2 = 47.5
Area = sqrt[47.5(47.5-25)(47.5-30)(47.5-40)] = sqrt[47.5×22.5×17.5×7.5]
= sqrt = 595.41 m²

Exercise 12.2 Solved Questions

Question 1: Equilateral ABC side 10cm. D,E,F midpoints. Find ar(ΔDEF).

Solution: ΔDEF = 1/4 ΔABC (midpoint theorem parallelograms)
ar(ABC) = (sqrt(3)/4) × 10² = 25sqrt(3)
ar(DEF) = (1/4) × 25sqrt(3) = 6.25sqrt(3) cm²

Question 2: ΔPQR sides 26cm, 28cm, 30cm. D midpoint QR. ar(ΔPDR)?

Solution: ar(PDR) = 1/2 ar(PQR) (same base height)
s = (26+28+30)/2 = 42
ar(PQR) = sqrt[42(42-26)(42-28)(42-30)] = sqrt[42×16×14×12] = 504 cm²
ar(PDR) = 252 cm²

Question 3: Field triangular ABC, heights 13cm, 14cm, 15cm from A,B,C. Sides?

Solution: Use area = (1/2)base×height for each → solve sides via Heron's

Question 4: Path widths 2m, 3m, 4m around equilateral field side 50m. Cost @ Rs 5/m²?

Solution: Outer triangle sides 54m, 56m, 58m → area outer - area field × 5

Question 5: ΔABC sides 25cm, 26cm, 27cm. D divides BC in 3:4. Areas ratio?

Solution: Same height → areas ratio BD:DC = 3:4

Question 6: Cost carpet Rs 20/m² for room triangular floor sides 15m,16m,17m?

Solution: s = 24, area = sqrt[24×9×8×7] = 120 m² × 20 = Rs 2400

Extra Practice Problems

Extra 1: Sides 9cm, 10cm, 17cm. Possible? No (9+10<17)
Extra 2: Sides 6cm, 8cm, 10cm. Area? 24 cm²
Extra 3: Equilateral 8cm side. Area? 16sqrt(3) cm²

15 FAQs

1. Heron's formula when? All three sides known

2. Semi-perimeter s? (a+b+c)/2

3. 5-12-13 triangle area? 30 cm²

4. Equilateral formula? (sqrt(3)/4)side²

5. Sides 7,8,10 area? ≈27.86 cm²

6. Midpoint triangle area? Half original

7. Triangle inequality fail? No triangle possible

8. Path around field? Outer - inner area

9. Units area? m², cm², etc.

10. Right triangle verify? (1/2)leg1×leg2

11. s(s-a) negative? Impossible triangle

12. Cost problems? Area × rate

13. Ratio areas same height? Base ratio

14. Field paths width? Outer parallel sides

15. Memory trick? s(s-a)(s-b)(s-c)

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