Mensuration

48 cm²

Step 1: Sides are a=10, b=10, c=12. s = (10+10+12)/2 = 32/2 = 16 cm. Step 2: s-a = 16-10 = 6, s-b = 16-10 = 6, s-c = 16-12 = 4. Step 3: Area = √[16×6×6×4] = √[2304] = 48 cm². Verification: Height h = √(10²-6²) = √(100-36) = √64 = 8 cm. Area = ½×12×8 = 48 cm². ✓

140 cm²

Step 1: A quadrilateral can be divided into two triangles by a diagonal. Here diagonal BD = 20 cm. Step 2: Area of triangle ABD = ½ × base × height = ½ × 20 × 8 = 80 cm². Step 3: Area of triangle BCD = ½ × 20 × 6 = 60 cm². Step 4: Area of quadrilateral ABCD = 80 + 60 = 140 cm². This is the standard method: divide quadrilateral into two triangles using a diagonal.

1200 m²

Step 1: Perimeter = 100 m, so each side = 100/4 = 25 m. One diagonal AC = 30 m, so half diagonal = 15 m. Step 2: For triangle ABC: a = 25, b = 25, c = 30. s = (25+25+30)/2 = 40 m. Step 3: Area of △ABC = √[40×(40-25)×(40-25)×(40-30)] = √[40×15×15×10] = √90000 = 300 m² (wait: 40×10=400, 15×15=225, 400×225=90000, √90000=300 m²). Step 4: Area of rhombus = 2 × 300 = 600... Let me recheck: √90000 = 300. Rhombus area = 2×300 = 600 m²? Actually checking again: s=40, s-25=15, s-25=15, s-30=10. Area=√(40×15×15×10)=√(90000)=300. Total = 2×300=600. The correct answer should be 600 m². However, the marked

2(lb + bh + lh)

Step 1: A cuboid has 6 rectangular faces: top, bottom, front, back, left, right. Step 2: Area of top and bottom = 2 × lb. Area of front and back = 2 × bh. Area of left and right = 2 × lh. Step 3: TSA = 2lb + 2bh + 2lh = 2(lb + bh + lh). Note: 2(l+b)h is LSA (Lateral Surface Area - only the 4 walls, excluding top and bottom). lbh is volume, not surface area.