Surface area of a combination of two solids
When two solids are joined, the touching faces are hidden. Add only the surfaces you can see, usually the curved surface areas of each part.
Why we do not add the two total surface areas
Think of an ice cream cone with a round scoop on top. The flat circle where the scoop sits on the cone is inside the solid. You cannot touch it or paint it. So the surface area of the new solid is the sum of the curved surface areas of the parts. It is not the sum of their total surface areas.
Common combinations
| Object | Parts | Surface area to find |
|---|---|---|
| Toy or lattu (top) | Cone on a hemisphere, same radius | πrl + 2πr² |
| Tent | Cone on a cylinder | 2πrh + πrl (canvas has no floor) |
| Medicine capsule | Cylinder with a hemisphere at each end | 2πrh + 2πr² + 2πr² = 2πrh + 4πr² |
| Rocket model with a closed base | Cone on a cylinder | πrl + 2πrh + πr² |
| Ice cream cone with a scoop | Hemisphere on a cone | πrl + 2πr² |
Worked example 1: a wooden toy
A wooden toy is a cone mounted on a hemisphere. Both have radius 7 cm. The total height of the toy is 31 cm. Find the total surface area of the toy. (Use π = 22/7.)
- Given: r = 7 cm, total height = 31 cm.
- The hemisphere is as tall as its radius, 7 cm. So the cone's height h = 31 − 7 = 24 cm.
- Slant height: l = √(r² + h²) = √(49 + 576) = √625 = 25 cm.
- Formula: surface area of toy = CSA of cone + CSA of hemisphere = πrl + 2πr².
- CSA of cone = (22/7) × 7 × 25 = 550 cm².
- CSA of hemisphere = 2 × (22/7) × 7 × 7 = 308 cm².
- Total = 550 + 308 = 858 cm².
Total surface area of the toy = 858 cm².
Worked example 2: canvas for a tent
A tent is a cylinder of radius 3.5 m and height 3 m with a conical top of the same radius. The slant height of the cone is 5 m. Find the canvas needed (there is no floor) and its cost at ₹ 200 per m². (Use π = 22/7.)
- Given: r = 3.5 m, cylinder height h = 3 m, slant height l = 5 m.
- Formula: canvas = CSA of cylinder + CSA of cone = 2πrh + πrl = πr(2h + l).
- πr = (22/7) × 3.5 = 11.
- Canvas = 11 × (2 × 3 + 5) = 11 × 11 = 121 m².
- Cost = 121 × ₹ 200 = ₹ 24,200.
Canvas needed = 121 m²; cost = ₹ 24,200.
How to draw the figure for a toy
- Draw a half circle (the hemisphere) with its flat side on top.
- On the flat side, draw a cone of the same width, with its tip pointing up.
- Draw the joining circle as a dotted ellipse: it is hidden inside.
- Mark r along the joining circle, h from the tip to the centre of the circle, and l along the side of the cone.
- Write the total height beside the figure, with an arrow from the tip to the bottom.
A Channapatna wooden top, a tent at a mela and a medicine capsule are all two solids joined. Makers work out only the outside area to know how much paint, canvas or coating to buy.
2 marks: “A medicine capsule is a cylinder of radius 3 mm and length 10 mm, with a hemisphere of the same radius at each end. Find its surface area in terms of π.” Surface area = 2πrh + 4πr² = 2π × 3 × 10 + 4π × 9 = 60π + 36π = 96π mm².
3 marks: “A rocket model is a cylinder of radius 3 cm and height 10 cm with a cone of height 4 cm on top. The base is closed. Find its total surface area. (Use π = 3.14.)” l = √(9 + 16) = 5 cm. TSA = πrl + 2πrh + πr² = π(15 + 60 + 9) = 84π = 84 × 3.14 = 263.76 cm².
5 marks: “A tent is a cylinder of radius 3 m and height 2 m with a cone of height 4 m on top. Find the canvas used (no floor) and its cost at ₹ 150 per m². (Use π = 3.14.)” l = √(9 + 16) = 5 m. Canvas = 2πrh + πrl = π(12 + 15) = 27 × 3.14 = 84.78 m². Cost = 84.78 × 150 = ₹ 12,717.
Joined faces hide. Count only what you can paint.