Heron's Formula

48 cm²

Step 1: Sides are a = 10 cm, b = 10 cm, c = 12 cm. Step 2: Calculate semi-perimeter s = (10+10+12)/2 = 16 cm. Step 3: Find differences: (s-a) = 6, (s-b) = 6, (s-c) = 4. Step 4: Apply Heron's formula: Area = √[16×6×6×4] = √[2304] = 48 cm². We can verify using height: height = √(10²-6²) = 8 cm, so Area = (1/2)×12×8 = 48 cm².

Right triangle, 84 cm²

Step 1: Check if it's a right triangle: 7² + 24² = 49 + 576 = 625 = 25². Yes, it's a right triangle. Step 2: For right triangle, area = (1/2) × 7 × 24 = 84 cm². Step 3: Verify using Heron's formula: s = (7+24+25)/2 = 28. Step 4: Area = √[28×21×4×3] = √[7056] = 84 cm². Both methods give the same answer, confirming our calculation.

60 cm²

Step 1: Check if it's a right triangle: 8² + 15² = 64 + 225 = 289 = 17². Yes, it's a right triangle. Step 2: For a right triangle, area = (1/2) × base × height = (1/2) × 8 × 15 = 60 cm². Step 3: Verify using Heron's formula: s = (8+15+17)/2 = 20. Step 4: Area = √[20×12×5×3] = √[3600] = 60 cm². This demonstrates how Heron's formula works even when we can use simpler methods.

216 m²

Step 1: Let sides be 5x, 12x, 13x. Since perimeter = 72, we have 30x = 72, so x = 2.4. Step 2: The sides are 12 m, 28.8 m, 31.2 m. Wait, let me recalculate: 5×2.4=12, 12×2.4=28.8, 13×2.4=31.2. Actually, 5+12+13=30, so 30x=72 gives x=2.4. Sides are 12m, 28.8m, 31.2m. Step 3: This is a 5:12:13 right triangle (5²+12²=13²). Step 4: Area = (1/2)×12×28.8 = 172.8 m². Let me recalculate: if ratio is 5:12:13 and perimeter is 72, then each unit = 72/30 = 2.4, giving sides 12, 28.8, 31.2. But 5:12:13 with scale factor 1.8 gives 9, 21.6, 23.4. Let me try again: 72/(5+12+13) = 72/30 = 2.4, so sides are 12,