Combined solids: surface area and volume
For a solid made of two or more shapes, add the volumes of the parts. For surface area, add only the outer surfaces; joined faces are hidden.
Rules for combined solids
- Volume: add the volumes of all the parts. If a part is cut out, subtract its volume.
- Surface area: add only the surfaces on the outside. Faces where two parts are joined are hidden, so leave them out.
- If a shape is scooped out, its inner surface is now on the outside, so add it.
- Write the dimensions of each part on your figure before you start.
Common combined solids
| Object | Parts | Total surface area | Volume |
|---|---|---|---|
| Toy | Cone on a hemisphere, same r | πrl + 2πr² | (1/3)πr²h + (2/3)πr³ |
| Medicine capsule | Cylinder with a hemisphere at each end | 2πrh + 4πr² | πr²h + (4/3)πr³ |
| Tent | Cylinder with a cone on top | 2πrh₁ + πrl (no floor) | πr²h₁ + (1/3)πr²h₂ (h₁ cylinder, h₂ cone) |
| Ice cream cone | Cone filled and topped with a hemisphere | πrl + 2πr² | (1/3)πr²h + (2/3)πr³ |
Example 1: a wooden toy
A wooden toy is a cone fixed on a hemisphere of the same radius, 7 cm. The total height of the toy is 31 cm. Find its total surface area and volume. (Take π = 22/7.)
- The height of the hemisphere is its radius, 7 cm. So the cone's height h = 31 − 7 = 24 cm.
- Slant height l = √(24² + 7²) = √625 = 25 cm
- TSA = CSA of cone + CSA of hemisphere = πrl + 2πr²
- = 22/7 × 7 × 25 + 2 × 22/7 × 7 × 7
- = 550 + 308 = 858 cm²
- Volume = (1/3)πr²h + (2/3)πr³ = (1/3)πr²(h + 2r)
- = 1/3 × 22/7 × 7 × 7 × (24 + 14) = 1/3 × 154 × 38
- = 5852/3 ≈ 1950.67 cm³
TSA = 858 cm², volume ≈ 1950.67 cm³
Example 2: a medicine capsule
A medicine capsule is a cylinder with a hemisphere at each end. It is 14 mm long and its radius is 3.5 mm. Find its surface area and volume. (Take π = 22/7.)
- Length of the cylinder part h = 14 − 3.5 − 3.5 = 7 mm.
- Surface area = CSA of cylinder + 2 × CSA of hemisphere = 2πrh + 4πr²
- = 2 × 22/7 × 3.5 × 7 + 4 × 22/7 × 3.5 × 3.5
- = 154 + 154 = 308 mm²
- Volume = πr²h + (4/3)πr³
- = 22/7 × 3.5 × 3.5 × 7 + 4/3 × 22/7 × 3.5 × 3.5 × 3.5
- = 269.5 + 179.67 = 449.17 mm³
Surface area = 308 mm², volume ≈ 449.17 mm³
How to draw a combined solid
- Draw each part with the same radius where they join, for example a cone standing on a semicircle.
- Draw the joining circle as a dotted ellipse, because it is hidden inside.
- Mark r, the height of each part and any slant height l.
- Write the total height and show how it splits, for example 31 cm = 24 cm + 7 cm.
A temple festival tent (cylinder + cone), a medicine capsule (cylinder + two hemispheres) and an ice cream cone (cone + hemisphere) are combined solids.
1 marks: “A solid is a cylinder with a hemisphere at each end. Which surfaces make up its surface area?” The curved surface of the cylinder and the curved surfaces of the two hemispheres: 2πrh + 4πr².
2 marks: “A tent is a cylinder of radius 7 m and height 4 m with a cone of slant height 10 m on top. Find the area of canvas needed. (π = 22/7)” Canvas = 2πrh + πrl = 2 × 22/7 × 7 × 4 + 22/7 × 7 × 10 = 176 + 220. Answer: 396 m².
5 marks: “A solid wooden cylinder has radius 5 cm and height 12 cm. A cone of the same radius and height is carved out from one end. Find the volume and total surface area of the remaining solid. (π = 3.14)” Volume = πr²h − (1/3)πr²h = 942 − 314 = 628 cm³. l = 13 cm. TSA = 2πrh + πr² + πrl = 376.8 + 78.5 + 204.1 = 659.4 cm².
Volumes all add up; surfaces count only if you can touch them.