01 · EXAM STRATEGY
Solving Complex Combined Shapes
Step 1
Break the composite object into standard geometric solids (Cylinder, Cone, Hemisphere, Cube).
Step 2
For Surface Area: Add exposed CSA of all outer parts. For Volume: Add (or subtract) volumes directly.
Step 3
Factor out common terms like πr or r² before calculating to avoid arithmetic mistakes!
02 · SOLVED NCERT EXAMPLES
Step-by-Step Problem Solving
EXAMPLE 1 · MEDICINE CAPSULE
A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and diameter is 5 mm. Find its surface area.
Diameter = 5 mm ⟹ Radius r = 2.5 mm.
Height of cylinder h = 14 - 2(2.5) = 14 - 5 = 9 mm.
Surface Area = CSA of Cylinder + 2 × (CSA of Hemisphere)
Surface Area = 2πrh + 2 × (2πr²) = 2πr (h + 2r)
Surface Area = 2 × (22/7) × 2.5 × (9 + 5)
Surface Area = 2 × (22/7) × 2.5 × 14 = 2 × 22 × 2.5 × 2 = 220 mm²
Surface Area = 2πrh + 2 × (2πr²) = 2πr (h + 2r)
Surface Area = 2 × (22/7) × 2.5 × (9 + 5)
Surface Area = 2 × (22/7) × 2.5 × 14 = 2 × 22 × 2.5 × 2 = 220 mm²
EXAMPLE 2 · TWO JOINED CUBES
2 cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid.
Volume of cube a³ = 64 cm³ ⟹ side a = 4 cm.
When two cubes are joined end to end:
Length of Cuboid l = 4 + 4 = 8 cm
Breadth b = 4 cm
Height h = 4 cm
Breadth b = 4 cm
Height h = 4 cm
Surface Area of Cuboid:
TSA = 2(lb + bh + hl)
TSA = 2[(8 × 4) + (4 × 4) + (4 × 8)]
TSA = 2(32 + 16 + 32) = 2(80) = 160 cm²
TSA = 2[(8 × 4) + (4 × 4) + (4 × 8)]
TSA = 2(32 + 16 + 32) = 2(80) = 160 cm²
03 · QUICK SUMMARY
Key Takeaways
01
Capsule surface area = 2πr (h + 2r).
02
When cubes are joined end to end, only the length doubles (breadth and height stay same!).
03
For pen stand conical depressions, subtract cone volumes from cuboid volume.