Solids — Important Questions

Volume of pyramid = (1/3) × base area × height, Volume of cone = (1/3)πr²h, Volume of sphere = (4/3)πr³, Volume of cylinder = πr²h

Step 1: Check each formula. Pyramid and cone volumes include the factor 1/3. Step 2: Sphere volume is (4/3)πr³. Step 3: Cylinder volume is πr²h (no factor of 1/3). The incorrect option omits the 1/3 factor for pyramid.

65π cm²

Step 1: Use formula for curved surface area of cone = πrl. Step 2: Substitute r = 5 cm, l = 13 cm. Step 3: Curved surface area = π × 5 × 13 = 65π cm². Common mistake: Using πr²l instead of πrl.

3√(3/π)^(1/3) cm

Step 1: Volume of cube = 6³ = 216 cm³. Step 2: Volume of sphere = (4/3)πr³. Step 3: Equating volumes: (4/3)πr³ = 216. Step 4: r³ = 216 × 3/(4π) = 162/π. Step 5: r = (162/π)^(1/3) = 3(6/π)^(1/3). Common mistake: Not equating volumes correctly.

All lateral faces are congruent isosceles triangles, The apex is directly above the center of the base, All lateral edges are equal, The base is a regular polygon

Step 1: Regular square pyramid has square base and congruent triangular faces. Step 2: Apex is centered above base. Step 3: All lateral edges from apex to base vertices are equal. Step 4: Base is square (regular polygon). The incorrect option states all faces are triangles, but the base is a square.