Paint only what you can see

The idea

The total surface area of a combined solid is the sum of the areas of the surfaces of its parts that remain exposed, not the sum of their total surface areas, because surfaces joined together disappear.

Gluing a cone on a hemisphere

Priya makes a round-bottomed toy by gluing a cone on a hemisphere of the same radius. Before gluing, the cone has 704 cm² of faces (curved face and base) and the hemisphere has 462 cm² (curved face and flat face).

How much paint (in cm² of surface) does the finished toy need?

• 1166 cm²
• 858 cm²
• 704 cm²

Join the solids, then paint

Slide the bar to join the solids. Then tap every face that you can still see, to paint it. Faces that are joined cannot be painted. Do this for all three solids.

Only the outside counts

The total surface area of a combined solid is the sum of the areas of the parts of the surface that stay exposed. It is not the sum of the total surface areas of the parts.

Where two solids are joined, the touching faces are hidden inside. For the toy: TSA = CSA of hemisphere + CSA of cone. With r = 7 cm and slant height 25 cm: 2πr² + πrl = 308 + 550 = 858 cm², not 462 + 704 = 1166 cm².

If only part of a face is covered, take away only the covered part. For a cube with a hemisphere on top: TSA = TSA of cube − πr² + 2πr², where πr² is the covered part of the top and 2πr² is the curved face of the hemisphere.

Notes

The total surface area of a combined solid is the sum of the areas of the parts of the surface that stay exposed, not the sum of the total surface areas of the parts.

Check yourself

A top is a cone with a hemisphere on it. The hemisphere has radius 1.75 cm. The cone has the same radius and slant height 3.7 cm. Find the area to be coloured. (Take π = 22/7)

Answer: 39.6 cm²

Hemisphere: 2πr² = 2 × 22/7 × 1.75 × 1.75 = 19.25. Cone: πrl = 22/7 × 1.75 × 3.7 = 20.35. Total = 39.6 cm².

A block is a cube of edge 5 cm with a hemisphere of diameter 4.2 cm fixed on top. Find its total surface area. (Take π = 22/7)

Answer: 163.86 cm²

150 − πr² + 2πr² = 150 + πr² = 150 + 22/7 × 2.1 × 2.1 = 150 + 13.86 = 163.86 cm².

A toy rocket is a cone on a cylinder. The cone has radius 2.5 cm and slant height 6.5 cm. The cylinder has radius 1.5 cm. The base of the cone is bigger than the cylinder, so a ring of it shows. Find the area of the cone part to paint orange: the curved face and the ring. (Take π = 3.14)

Answer: 63.585 cm²

πrl + πr² − π(r′)² = 3.14 × (2.5 × 6.5 + 2.5² − 1.5²) = 3.14 × 20.25 = 63.585 cm².

A toy is made by joining a cone and a hemisphere of the same radius at their flat faces. What is its total surface area?

• CSA of the cone + CSA of the hemisphere — correct. Yes! The joined circles are inside, so only the two curved faces stay outside.
• TSA of the cone + TSA of the hemisphere. That also counts the two joined circles (the base of the cone and the flat face of the hemisphere), which are hidden.
• CSA of the cone + TSA of the hemisphere. The flat face of the hemisphere is joined to the cone, so it is hidden. Only its curved face counts.