Surface Areas and Volumes
Surface area of combined solids
Find the surface area of a solid made of two or more shapes.
A water tanker on a truck is a cylinder with a hemisphere at each end. A test tube is a cylinder with one hemisphere. Many objects are two or more basic solids joined together.
Surface area = every part you can touch from outside. A quick reminder:
| Solid | Curved surface | Total surface |
|---|---|---|
| cylinder | 2πrh | 2πr(h + r) |
| cone (slant l = √(h² + r²)) | πrl | πr(l + r) |
| sphere | 4πr² | 4πr² (no flat face) |
| hemisphere | 2πr² | 3πr² (dome + flat circle) |
When two solids are joined, the faces where they meet disappear: you can't touch them any more. So the total is not the two total surface areas added together.
A routine that works:
- start with the first solid's surface
- add the parts of the second solid that show
- subtract any face that got covered
Example: a hemisphere stuck on top of a cube. The cube loses a circle on top (–πr²) but gains the dome (+2πr²). Overall it adds +πr².
Example: a top shaped like a cone on a hemisphere. Only the two curved surfaces show: 2πr² + πrl.
Watch what you can actually see. A toy rocket has a cone of diameter 5 cm on a cylinder of diameter 3 cm. The cone is wider, so a ring of its base still shows and must be painted: π(2.5² – 1.5²) of extra area.
Heights in combined solids: a hemisphere's height is its radius. So a cone on a hemisphere, 15.5 cm tall with r = 3.5 cm, has a cone 12 cm high.
Hollow means you can put your hand inside: count the inside surface, and leave out any lid that isn't there.
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