What the join hides
Add every face of the toy. The total is more than the paint needs.
- Two faces meet at the rim.
- The seam is never on the outside.
- One disc belongs to the cone.
- One disc belongs to the dome.
The circles that touch
The toy has four faces. The cone has a curved skin and a flat base. The dome has a curved skin and a flat face.
- Cone skin 2310, cone base 1386.
- Dome skin 2772, flat face 1386.
- All four added: 7854.
- Only the outside: 2310 + 2772 = 5082.
The gap is 7854 − 5082 = 2772. That is two circles of 1386. The cone's base and the dome's flat face touch each other. Both are inside.
The touching part is hidden, and only that part. Take a cube with a hemisphere fixed on its top. The hemisphere covers a circle of the top face. The rest of the top face still shows.
Take cube edge 10 and hemisphere radius 3.5. Here r = 3.5 is the hemisphere's radius. The cube has 6 faces of 10 × 10 = 100. The flat face of the hemisphere covers πr² of the top. Its dome adds 2πr². So the surface is 6 × 100 − πr² + 2πr². Here πr² = 22/7 × 3.5 × 3.5 = 38.5. That is 600 + 38.5 = 638.5.
A tank is a drum with a dome fixed on its top. The bottom disc is outside, so it counts. Only the drum's top disc and the dome's flat face touch.
Where two solids join, the touching parts are inside. Do not count them in the outside area. Count every part you can see.
Decide which faces count.
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