The surface area of a compound 3D shape is the total of all exposed faces — the faces you could paint if you dipped the whole solid in paint. Any faces hidden inside the join are covered and must be subtracted from each part's full surface area before adding the parts together.
What is a compound 3D shape?
A compound 3D solid is formed by joining two or more standard solids (such as a cylinder, cone, prism, hemisphere or pyramid) together. Where two solids meet, their shared circular or polygonal faces become internal and are no longer part of the surface.
Key principle: Identify every face of each component solid, then remove any face that has been joined to another solid (hidden faces). Add up only the remaining exposed faces.
How do you find the surface area of a cylinder topped with a hemisphere?
Example: A solid is formed by placing a hemisphere (radius 3 cm) on top of a cylinder (radius 3 cm, height 8 cm). Find the total surface area. Leave your answer in terms of π.
Identify the faces of each part:
| Part | Faces | Include? |
|---|---|---|
| Cylinder | Bottom circle: π × 3² = 9π | ✓ Exposed |
| Cylinder | Curved surface: 2π × 3 × 8 = 48π | ✓ Exposed |
| Cylinder | Top circle: π × 3² = 9π | ✗ Hidden (hemisphere sits here) |
| Hemisphere | Curved surface: 2π × 3² = 18π | ✓ Exposed |
| Hemisphere | Flat circular base: π × 3² = 9π | ✗ Hidden (sits on cylinder top) |
Total surface area = 9π + 48π + 18π = 75π cm²
Note: the two hidden faces (cylinder top and hemisphere base) are equal circles, so they always cancel each other out when one solid is placed flat on the other.
How do you find the surface area of a cone on a cylinder?
Example: A cone (radius 4 cm, slant height 6 cm) sits on top of a cylinder (radius 4 cm, height 10 cm). Find the total surface area. Use π ≈ 3.14.
Formulae needed:
- Curved surface of cone = πrl (r = radius, l = slant height)
- Curved surface of cylinder = 2πrh
- Circle area = πr²
| Part | Calculation | Value (cm²) |
|---|---|---|
| Cylinder bottom circle | π × 4² | 50.24 |
| Cylinder curved surface | 2π × 4 × 10 | 251.20 |
| Cylinder top circle | Hidden — cone sits here | — |
| Cone curved surface | π × 4 × 6 | 75.36 |
| Cone base circle | Hidden — sits on cylinder | — |
Total = 50.24 + 251.20 + 75.36 = 376.80 cm²
(Using π = 3.14.)
How do you find the surface area of a prism with a pyramid on top?
Example: A square-based pyramid (base 6 cm × 6 cm, slant height 5 cm) sits on top of a cuboid (6 cm × 6 cm × 4 cm). Find the total surface area.
Cuboid faces:
- Two 6 × 6 faces (top and bottom): 2 × 36 = 72 cm²
- Four 6 × 4 faces: 4 × 24 = 96 cm²
- Total cuboid surface area = 168 cm²
- Subtract the hidden top face: −36 cm²
- Cuboid contribution: 132 cm²
Pyramid faces:
- Four triangular faces: 4 × (½ × 6 × 5) = 4 × 15 = 60 cm²
- Base is hidden: not included.
- Pyramid contribution: 60 cm²
Total surface area = 132 + 60 = 192 cm²
How do you organise your working to avoid missing faces?
Use a systematic table or checklist:
- List every face of Part 1.
- List every face of Part 2.
- Identify the joined (hidden) faces and remove them from both lists.
- Calculate the area of each remaining face.
- Sum all the remaining areas.
Tip: The hidden faces at the join are always identical (same shape and dimensions), one from each solid, so they come in matching pairs to remove.
What mistakes should you avoid?
Mistake 1 — Including the joined faces. This is the most common error. When a hemisphere sits on a cylinder, neither the top of the cylinder nor the flat base of the hemisphere is exposed; both must be excluded.
Mistake 2 — Using the diameter instead of the radius. The area and circumference formulae for circles and cylinders require the radius r. If the question gives a diameter d, halve it: r = d/2.
Mistake 3 — Using the perpendicular height of a cone instead of the slant height. The curved surface area of a cone is πrl where l is the slant height, not the vertical height h. Use Pythagoras to find l if you are given h: l = √(r² + h²).
Frequently asked questions
How do I find the slant height of a cone if I am given only the vertical height and radius?
Apply Pythagoras' theorem. In the right-angled triangle formed by the vertical height h, the radius r and the slant height l: l² = r² + h². So l = √(r² + h²). For a cone with r = 3 cm and h = 4 cm: l = √(9 + 16) = √25 = 5 cm.
What is the curved surface area of a hemisphere?
The curved surface area of a full sphere is 4πr². A hemisphere is half a sphere, so its curved surface is 2πr². Its flat circular base has area πr², but this is excluded when the hemisphere is joined to another solid.
Does the order of the shapes in a compound solid matter?
The total surface area depends on which faces are hidden, not on which shape is "on top". Two shapes joined with the same flat circular face will always have the same hidden area subtracted, regardless of which way up the solid is drawn.
How do I check my surface area answer is reasonable?
Find the surface area of each part in isolation (including all faces). The compound surface area must be less than the sum of those individual surface areas, because at least two faces are hidden. If your answer equals or exceeds the sum of the individual areas, you have forgotten to remove the joined faces.
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