Flat and curved composite surfaces
B is the complete cuboid area, T the complete cylinder area and C one contact circle. S is the outside area after joining
- Identify the contact For a cylinder sitting wholly on a cuboid, each touching patch has area pi times the radius squared
- Check exclusions Subtract the cuboid underside once only when the task excludes it
- Keep exact values Convert lengths first, retain π and round the final area
Worked example
A cylinder with radius 2 cm and height 3 cm is attached to an 8 cm by 6 cm by 2 cm cuboid. Find the painted area when the cuboid underside is excluded
- \(\displaystyle B=2(48+16+12)=152\)Complete cuboid area, in cm²
- \(\displaystyle T=8\pi+12\pi=20\pi\)Complete cylinder area, in cm²
- \(\displaystyle S=152+20\pi-8\pi\)Remove both internal circular contact surfaces
- \(\displaystyle A=104+12\pi\)Also subtract the 48 cm² cuboid underside
- \(\displaystyle A\approx141.70\text{ cm}^2\)Round only the final value to 2 decimal places
Common mistakes
Difficulty levels
Questions about this topic
Why can the result simplify to cuboid area plus cylinder wall?+
The exposed cylinder top has exactly the same area as the covered cuboid patch. Their areas replace each other. The cylinder base is internal
Does that simplification work for every joined solid?+
No. It uses the matching circular top and base of this cylinder, a whole base in contact, and an exposed cylinder top. Otherwise account for every included surface separately
Should I subtract the underside automatically?+
No. Read whether the task includes or excludes it
Why not round each separate area?+
Early rounding can change the final result. Keep exact expressions involving π until all additions and subtractions are complete
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 1: Measurement