Heron S Formula — Karnataka (SSLC) Class 9 Mathematics Solutions (Free)

Step 1: Heron's Formula: For a triangle with sides a, b, and c: Semi-perimeter s = (a + b + c)/2 Area = √[s(s - a)(s - b)(s - c)] This formula is used when the height is not given but all three sides are known. Step 2: Given sides: a = 5 cm b = 6 cm c = 7 cm Step 3: First, verify that these sides can form a triangle using triangle inequality: 5 + 6 = 11 > 7 ✓ 6 + 7 = 13 > 5 ✓ 5 + 7 = 12 > 6 ✓ Step 4: Calculate semi-perimeter: s = (a + b + c)/2 s = (5 + 6 + 7)/2 s = 18/2 s = 9 cm Step 5: Calculate (s - a), (s - b), (s - c): s - a = 9 - 5 = 4 s - b = 9 - 6 = 3 s - c = 9 - 7 = 2 Step 6: Apply Heron's formula: Area = √[s(s - a)(s - b)(s - c)] Area = √[9 × 4 × 3 × 2] Area = √[9 × 24] Area = √216 Step 7: Simplify √216: 216 = 36 × 6 √216 = √(36 × 6) = 6√6 cm² Step 8: Numerical value: √6 ≈ 2.449 Area ≈ 6 × 2.449 ≈ 14.70 cm² Answer: Area of the triangle = 6√6 cm² ≈ 14.70 cm².

Step 1: Given sides of isosceles triangle: a = 10 cm b = 10 cm c = 12 cm Step 2: Verify triangle inequality: 10 + 10 = 20 > 12 ✓ 10 + 12 = 22 > 10 ✓ 10 + 12 = 22 > 10 ✓ Step 3: Calculate semi-perimeter: s = (a + b + c)/2 s = (10 + 10 + 12)/2 s = 32/2 s = 16 cm Step 4: Calculate the differences: s - a = 16 - 10 = 6 s - b = 16 - 10 = 6 s - c = 16 - 12 = 4 Step 5: Apply Heron's formula: Area = √[s(s - a)(s - b)(s - c)] Area = √[16 × 6 × 6 × 4] Area = √[16 × 36 × 4] Area = √[16 × 144] Area = √2304 Area = 48 cm² Step 6: Verification using base-height method: For isosceles triangle with equal sides 10 cm and base 12 cm: Draw altitude from apex to base, which bisects the base (6 cm each). In right triangle: 10² = 6² + h² 100 = 36 + h² h² = 64 h = 8 cm Area = (1/2) × base × height = (1/2) × 12 × 8 = 48 cm² ✓ Answer: Area of the isosceles triangle = 48 cm².

Step 1: Given sides: a = 13 cm b = 14 cm c = 15 cm Step 2: Verify triangle inequality: 13 + 14 = 27 > 15 ✓ 14 + 15 = 29 > 13 ✓ 13 + 15 = 28 > 14 ✓ Step 3: Calculate semi-perimeter: s = (a + b + c)/2 s = (13 + 14 + 15)/2 s = 42/2 s = 21 cm Step 4: Calculate the differences: s - a = 21 - 13 = 8 s - b = 21 - 14 = 7 s - c = 21 - 15 = 6 Step 5: Apply Heron's formula: Area = √[s(s - a)(s - b)(s - c)] Area = √[21 × 8 × 7 × 6] Step 6: Calculate the product: 21 × 8 = 168 7 × 6 = 42 168 × 42 = 7056 Area = √7056 Step 7: Factorize 7056: 7056 = 16 × 441 = 16 × 441 441 = 21² 7056 = 16 × 21² √7056 = √(16 × 441) = 4 × 21 = 84 cm² Step 8: Verification by another method: 7056 = 2⁴ × 3² × 7² √7056 = 2² × 3 × 7 = 4 × 3 × 7 = 84 cm² Answer: Area of the triangle = 84 cm².

Step 1: Given: Sides in ratio 3:4:5, Perimeter = 36 cm Step 2: Let the sides be 3x, 4x, and 5x (where x is a constant). Step 3: Use perimeter to find x: Perimeter = 3x + 4x + 5x = 36 12x = 36 x = 3 Step 4: Find the actual sides: Side a = 3x = 3 × 3 = 9 cm Side b = 4x = 4 × 3 = 12 cm Side c = 5x = 5 × 3 = 15 cm Step 5: Verify: 9 + 12 + 15 = 36 ✓ Step 6: Note that 9, 12, 15 are multiples of 3, 4, 5, which is a Pythagorean triplet. Verification: 9² + 12² = 81 + 144 = 225 = 15² This is a right triangle with legs 9 cm and 12 cm, and hypotenuse 15 cm. Step 7: For a right triangle, area can be calculated as: Area = (1/2) × leg₁ × leg₂ = (1/2) × 9 × 12 = 54 cm² Step 8: Verify using Heron's formula: s = (9 + 12 + 15)/2 = 36/2 = 18 cm Area = √[18 × (18-9) × (18-12) × (18-15)] Area = √[18 × 9 × 6 × 3] Area = √[18 × 9 × 18] Area = √[18² × 9] Area = 18√9 Area = 18 × 3 Area = 54 cm² ✓ Answer: Area of the triangle = 54 cm².

Step 1: Given sides: a = 7 cm b = 8 cm c = 9 cm Step 2: Verify triangle inequality: 7 + 8 = 15 > 9 ✓ 8 + 9 = 17 > 7 ✓ 7 + 9 = 16 > 8 ✓ Step 3: Calculate semi-perimeter: s = (a + b + c)/2 s = (7 + 8 + 9)/2 s = 24/2 s = 12 cm Step 4: Calculate the differences: s - a = 12 - 7 = 5 s - b = 12 - 8 = 4 s - c = 12 - 9 = 3 Step 5: Apply Heron's formula: Area = √[s(s - a)(s - b)(s - c)] Area = √[12 × 5 × 4 × 3] Area = √[12 × 60] Area = √720 Step 6: Simplify √720: 720 = 144 × 5 = 12² × 5 √720 = √(144 × 5) = 12√5 cm² Step 7: Calculate decimal value: √5 ≈ 2.236 Area ≈ 12 × 2.236 ≈ 26.83 cm² Step 8: Verification by factoring: 720 = 16 × 45 = 16 × 9 × 5 = 144 × 5 √720 = √(144 × 5) = 12√5 ✓ Answer: Area = 12√5 cm² ≈ 26.83 cm².

Step 1: For an equilateral triangle with side a: All sides are equal: sides = a, a, a Step 2: Calculate semi-perimeter: s = (a + a + a)/2 = 3a/2 Step 3: Calculate the differences: s - a = 3a/2 - a = a/2 s - a = 3a/2 - a = a/2 s - a = 3a/2 - a = a/2 All three differences equal a/2 (property of equilateral triangles). Step 4: Apply Heron's formula: Area = √[s(s - a)(s - a)(s - a)] Area = √[(3a/2) × (a/2) × (a/2) × (a/2)] Area = √[(3a/2) × (a/2)³] Area = √[(3a/2) × (a³/8)] Area = √[3a⁴/16] Area = √(3a⁴)/√16 Area = a²√3/4 cm² Step 5: Compare with standard formula: Standard formula: Area = (√3/4)a² Heron's formula result: Area = (√3/4)a² Both are identical ✓ Step 6: Example with a = 6 cm: Area = (√3/4) × 6² Area = (√3/4) × 36 Area = 9√3 cm² Area ≈ 9 × 1.732 ≈ 15.59 cm² Answer: Area of equilateral triangle with side a = (√3/4)a² = a²√3/4 cm². Formula derived from Heron's formula matches the standard formula.

Step 1: Given sides of triangular field: a = 50 m b = 60 m c = 70 m Step 2: Verify triangle inequality: 50 + 60 = 110 > 70 ✓ 60 + 70 = 130 > 50 ✓ 50 + 70 = 120 > 60 ✓ Step 3: Calculate semi-perimeter: s = (a + b + c)/2 s = (50 + 60 + 70)/2 s = 180/2 s = 90 m Step 4: Calculate the differences: s - a = 90 - 50 = 40 s - b = 90 - 60 = 30 s - c = 90 - 70 = 20 Step 5: Apply Heron's formula: Area = √[s(s - a)(s - b)(s - c)] Area = √[90 × 40 × 30 × 20] Area = √[90 × 40 × 30 × 20] Step 6: Calculate the product: 90 × 40 = 3600 30 × 20 = 600 3600 × 600 = 2,160,000 Area = √2,160,000 Step 7: Simplify: 2,160,000 = 21,600 × 100 = 216 × 10,000 √2,160,000 = √(216 × 10,000) = 100√216 = 100 × 6√6 = 600√6 m² Step 8: Numerical value: √6 ≈ 2.449 Area ≈ 600 × 2.449 ≈ 1469.4 m² Step 9: Calculate perimeter (for fencing): Perimeter = a + b + c = 50 + 60 + 70 = 180 m Step 10: Cost of fencing: Cost per meter = Rs 5 Total cost = Perimeter × Cost per meter Total cost = 180 × 5 = Rs 900 Answer: Area of the triangular field = 600√6 m² ≈ 1469.4 m²; Total cost of fencing = Rs 900.

Step 1: Right triangle with sides: a = 3 cm (one leg) b = 4 cm (other leg) c = 5 cm (hypotenuse) Step 2: Verify it's a right triangle: 3² + 4² = 9 + 16 = 25 = 5² ✓ This is a Pythagorean triplet. Step 3: Using standard right triangle formula: Area = (1/2) × base × height Area = (1/2) × 3 × 4 Area = (1/2) × 12 Area = 6 cm² Step 4: Using Heron's formula: Calculate semi-perimeter: s = (3 + 4 + 5)/2 s = 12/2 s = 6 cm Step 5: Calculate the differences: s - a = 6 - 3 = 3 s - b = 6 - 4 = 2 s - c = 6 - 5 = 1 Step 6: Apply Heron's formula: Area = √[s(s - a)(s - b)(s - c)] Area = √[6 × 3 × 2 × 1] Area = √36 Area = 6 cm² Step 7: Comparison: Standard formula result: 6 cm² Heron's formula result: 6 cm² Both methods give identical result ✓ Step 8: This demonstrates that Heron's formula is a general formula that works for all triangles, including right triangles, and produces the same result as the specific formula for right triangles. Answer: Area using Heron's formula = 6 cm²; Area using right triangle formula = 6 cm². Both methods confirm the area is 6 cm².