Surface area of combination of solids — cone on cylinder, hemisphere on cone

Question

How do we find the total surface area of a solid formed by combining basic shapes like cones, cylinders, and hemispheres?

Solution — Step by Step

When two solids are joined, the faces that touch each other become internal and are no longer part of the surface. So:

\text{TSA of combination} = \text{Sum of CSAs of all parts} + \text{exposed flat faces} - \text{joined faces}

In practice, we add up the curved surface areas (CSAs) of each part and include only those flat faces that remain exposed.

A cone (radius $r$ , slant height $l$ ) sits on top of a cylinder (radius $r$ , height $h$ ):

CSA of cylinder = $2\pi r h$
CSA of cone = $\pi r l$
Bottom of cylinder = $\pi r^2$ (exposed)
Top of cylinder = hidden (cone sits on it)
Base of cone = hidden (same circle as top of cylinder)

\text{TSA} = 2\pi r h + \pi r l + \pi r^2

A hemisphere (radius $r$ ) sits on a cylinder (radius $r$ , height $h$ ):

CSA of cylinder = $2\pi r h$
CSA of hemisphere = $2\pi r^2$
Bottom of cylinder = $\pi r^2$ (exposed)
The flat face of hemisphere and top of cylinder cancel out

\text{TSA} = 2\pi r h + 2\pi r^2 + \pi r^2 = 2\pi r h + 3\pi r^2

Identify each basic solid in the combination
List all surfaces of each solid (curved + flat)
Mark which flat faces are joined (internal) — these are removed
Add remaining curved surface areas and exposed flat faces

For solids of revolution (like a capsule shape = cylinder + 2 hemispheres), there are often no exposed flat faces at all. The total surface area is just the sum of curved surface areas. A medicine capsule has TSA = $2\pi rh + 2 \times 2\pi r^2 = 2\pi r(h + 2r)$ .

flowchart TD
    A["Combination of Solids: Find TSA"] --> B["Identify each basic solid"]
    B --> C["List all surfaces: curved + flat"]
    C --> D["Identify joined flat faces"]
    D --> E["Remove joined faces from the total"]
    E --> F["TSA = Sum of CSAs + exposed flat faces"]

Why This Works

Surface area is the total area of the outer boundary. When solids are combined, some previously external faces become internal boundaries. We must exclude these from the total because they are no longer part of the outer surface. Only the parts visible from outside contribute to the total surface area.

Alternative Method

For complex combinations, build the answer incrementally. Start with the full TSA of the largest solid, then for each attached piece: add its CSA and subtract the area of the joint. This “add CSA, subtract joint” rule handles any number of attached pieces.

Common Mistake

Students add up the TSAs of individual solids instead of the CSAs. If you add TSA of a cone ( $\pi rl + \pi r^2$ ) and TSA of a cylinder ( $2\pi rh + 2\pi r^2$ ), you are double-counting the joined circular face and including it instead of removing it. Always work with CSAs and then carefully add only the exposed flat faces. This error appears in CBSE 10th boards every year and costs 2-3 marks.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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