Surface area when a part is stuck on or cut out
A hemisphere stuck on a flat face hides a circle πr² but adds 2πr². So the area goes up by πr². Scooping a hemisphere out also adds πr².
Subtract what goes, add what comes
When a hemisphere is stuck on top of a cube, a circle on the cube's top is covered. You subtract that circle, πr². The hemisphere adds its curved dome, 2πr². When a hemisphere is scooped out, the circle is removed, but a bowl-shaped inner surface appears. Again you subtract πr² and add 2πr².
Stuck on or cut out: what changes
| Situation | Subtract | Add | Net change |
|---|---|---|---|
| Hemisphere stuck on a face of a cube | πr² (covered circle) | 2πr² (curved dome) | + πr² |
| Hemisphere scooped out of a face | πr² (removed circle) | 2πr² (inner bowl) | + πr² |
| Cone cut out of one end of a cylinder (same radius) | πr² (top circle) | πrl (inner cone) | + πrl − πr² |
| Hemisphere scooped from each end of a cylinder (same radius) | 2πr² (both end circles) | 4πr² (two inner bowls) | + 2πr² |
Worked example 1: a decorative block
A cube of edge 14 cm has a hemisphere of diameter 7 cm fixed on its top face. Find the total surface area of the block. (Use π = 22/7.)
- Given: edge a = 14 cm; hemisphere radius r = 7 ÷ 2 = 3.5 cm.
- TSA of cube = 6a² = 6 × 14 × 14 = 1176 cm².
- Circle covered = πr² = (22/7) × 3.5 × 3.5 = 38.5 cm².
- CSA of hemisphere = 2πr² = 2 × 38.5 = 77 cm².
- TSA of block = 1176 − 38.5 + 77 = 1214.5 cm².
Total surface area of the block = 1214.5 cm².
Worked example 2: a wooden article
A wooden cylinder is 12 cm high with radius 3.5 cm. A hemisphere of the same radius is scooped out of each end. Find the total surface area of the article. (Use π = 22/7.)
- Given: r = 3.5 cm, h = 12 cm.
- Both flat ends are fully removed, because each hemisphere has the same radius as the cylinder.
- Surfaces left: CSA of cylinder + inner CSA of two hemispheres.
- Formula: TSA = 2πrh + 2 × 2πr² = 2πr(h + 2r).
- 2πr = 2 × (22/7) × 3.5 = 22.
- TSA = 22 × (12 + 7) = 22 × 19 = 418 cm².
Total surface area of the article = 418 cm².
Worked example 3: a cone cut out of a cylinder
From a solid cylinder of radius 3 cm and height 4 cm, a cone of the same radius and height is hollowed out. Find the total surface area of the remaining solid. (Use π = 3.14.)
- Given: r = 3 cm, h = 4 cm. Slant height of the cone l = √(9 + 16) = 5 cm.
- Surfaces left: CSA of cylinder + bottom circle + inner CSA of cone.
- Formula: TSA = 2πrh + πr² + πrl = π(2 × 3 × 4 + 3 × 3 + 3 × 5).
- TSA = π(24 + 9 + 15) = 48π = 48 × 3.14 = 150.72 cm².
Total surface area of the remaining solid = 150.72 cm².
A brass trophy with a dome on a square base, or a wooden kumkum box with a bowl scooped in, is polished on every visible face. The polishing cost depends on all these faces, including the new inner surface.
1 marks: “A hemisphere of radius r is placed on a face of a cube. By how much does the surface area increase?” By πr²: the dome adds 2πr² and covers πr².
3 marks: “A hemispherical hollow of diameter 10 cm is scooped out of one face of a cube of edge 10 cm. Find the total surface area of the remaining solid. (Use π = 3.14.)” r = 5 cm. TSA = 6a² − πr² + 2πr² = 600 + πr² = 600 + 78.5 = 678.5 cm².
3 marks: “From a solid cylinder of height 8 cm and radius 6 cm, a cone of the same height and radius is hollowed out. Find the total surface area of the remaining solid in terms of π.” l = √(36 + 64) = 10 cm. TSA = 2πrh + πr² + πrl = π(96 + 36 + 60) = 192π cm².
Stuck on or scooped out, a hemisphere always adds πr² to the surface area.