Combined solids

A combined solid is built by attaching two or more basic shapes , a cone on top of a cylinder, a hemisphere capping a cylinder, two hemispheres glued together, and so on. Real objects often have this composite structure: a tent (cone + cylinder), a candle (cone + cylinder), a capsule (cylinder + 2 hemispheres), a toy (hemisphere + cone), a water tanker (cylinder + 2 hemispheres or 2 cones).

The two essential rules for combined solids:

1. Volumes add up directly. The combined volume is the sum of the individual volumes.
2. Surface area adds the external surfaces only. Hidden joining surfaces are excluded.

The second rule is the trickier one. When you join a cone to a cylinder, the cone's base disk and the cylinder's top disk coincide , they touch each other and are inside the solid, so both should be excluded.

Strategy

1. Identify the basic shapes that make up the combined solid.
2. List the surfaces of each basic shape: curved surfaces, base discs, etc.
3. Mark the hidden surfaces (where the shapes touch). Exclude them.
4. Compute the external surface area by summing what remains.
5. Compute the volume by summing the basic volumes.

Standard combinations

Cone on cylinder (e.g., a tent). Cylinder radius = cone radius = $r$. Cylinder height $h_1$, cone height $h_2$, slant $\ell$. External surface = cylinder's CSA + cone's CSA + cylinder's bottom disk (base of the tent):

$S = 2\pi r h_1 + \pi r \ell + \pi r^2.$

Volume:

$V = \pi r^2 h_1 + (1/3) \pi r^2 h_2.$

Hemisphere on cylinder (e.g., a granary). External surface = cylinder's CSA + hemisphere's curved area + cylinder's bottom disk:

$S = 2\pi r h + 2\pi r^2 + \pi r^2 = 2\pi r h + 3 \pi r^2.$

Volume:

$V = \pi r^2 h + (2/3)\pi r^3.$

Hemisphere on cone (e.g., a toy: spinning top). External surface = hemisphere's curved area + cone's CSA:

$S = 2\pi r^2 + \pi r \ell.$

Volume:

$V = (2/3) \pi r^3 + (1/3) \pi r^2 h.$

Two hemispheres + cylinder (e.g., a capsule). External surface = cylinder's CSA + two hemispheres' curved areas:

$S = 2\pi r h + 2 \cdot 2\pi r^2 = 2\pi r h + 4\pi r^2.$

Volume:

$V = \pi r^2 h + 2 \cdot (2/3) \pi r^3 = \pi r^2 h + (4/3) \pi r^3.$

(Note: the two hemispheres together make a sphere, so this is cylinder + sphere.)

Cone hollowed from a cylinder. A cone is removed from a cylinder of the same radius and same height. The remaining external area = cylinder TSA + cone CSA + one circular top disc minus... actually, this depends on whether the cone is inverted (apex at the bottom of the cylinder). Sketch the geometry carefully.

Hemisphere hollowed from a cone (e.g., an ice-cream cone with a hemispherical scoop sticking up). The ice cream is the hemisphere + the cone, so volume is sum.

Worked examples

Example 1. A toy is made by joining a cone of height $4$ cm and radius $3$ cm to a hemisphere of the same radius. Find the total surface area.

Slant height of cone: $\ell = \sqrt{9 + 16} = 5$.

External surface = cone's CSA + hemisphere's curved area $= \pi \cdot 3 \cdot 5 + 2\pi \cdot 9 = 15\pi + 18\pi = 33\pi \approx 103.71$ cm$^2$.

Example 2. A medical capsule consists of a cylinder of length $10$ mm with diameter $5$ mm, capped by hemispheres at both ends. Find the surface area.

$r = 2.5$, $h = 10 - 2 \cdot 2.5 = 5$ (the cylinder's length excluding hemispheres, if the total length is $10$).

Surface area = $2\pi r h + 4\pi r^2 = 2\pi(2.5)(5) + 4\pi(6.25) = 25\pi + 25\pi = 50\pi \approx 157$ mm$^2$.

Example 3. A solid is in the form of a cylinder with hemispherical ends. The total length is $35$ cm and the diameter is $14$ cm. Find the total surface area.

$r = 7$. Length of cylinder excluding hemispheres $= 35 - 14 = 21$.

$S = 2\pi r h + 4\pi r^2 = 2 \cdot 22/7 \cdot 7 \cdot 21 + 4 \cdot 22/7 \cdot 49 = 924 + 616 = 1540$ cm$^2$.

Example 4. A cone of height $14$ cm is placed on top of a cylinder of height $14$ cm. Both have radius $7$ cm. Find the total volume.

$V = \pi r^2 h_{\text{cyl}} + (1/3) \pi r^2 h_{\text{cone}} = 22/7 \cdot 49 \cdot 14 + (1/3) \cdot 22/7 \cdot 49 \cdot 14 = 2156 + 718.67 = 2874.67$ cm$^3$.

Example 5. A vessel is in the form of a hemispherical bowl mounted by a hollow cylinder. The inner radius is $7$ cm and the cylinder's height is $5$ cm. Find the capacity.

$V = (2/3)\pi r^3 + \pi r^2 h = (2/3)(22/7)(343) + (22/7)(49)(5) = 718.67 + 770 = 1488.67$ cm$^3$.

Example 6. A wooden article is made by scooping a hemisphere out of one end of a solid cylinder of height $10$ cm and radius $3.5$ cm. Find the total surface area.

External area = cylinder's CSA + one circular end (the un-scooped one) + hemisphere's curved area (since the hemisphere is hollowed inward from the other end, you see its inner curved surface):

$S = 2\pi r h + \pi r^2 + 2\pi r^2 = 2\pi r h + 3\pi r^2 = 2 \cdot 22/7 \cdot 3.5 \cdot 10 + 3 \cdot 22/7 \cdot 12.25 = 220 + 115.5 = 335.5$ cm$^2$.

Try it yourself

1. A tent is a cylinder of radius $4$ m and height $3$ m, topped by a cone of slant height $5$ m. Find canvas required (CSA of both, plus base if needed).
2. A child's toy is a cone on a hemisphere, both with radius $7$ cm. The cone's height is $14$ cm. Find total surface area.
3. A solid sphere of radius $3$ is melted with a cone of radius $3$ and height $5$ to recast a cylinder of radius $3$. Find its height.
4. A capsule has cylinder length $10$ mm and diameter $4$ mm, hemispherical ends. Find total volume.
5. A jar is cylindrical of height $20$ cm and radius $7$ cm, with a hemispherical base. Find the capacity.
6. A cone is hollowed from a cylinder of radius $3$ and height $7$, same radius and height. Find remaining volume.
7. A hemisphere of radius $4$ cm is mounted on a cone of slant height $5$ cm and same radius. Find total surface area.
8. A bucket holds $5$ litres of water. If it is in the form of a cylinder of radius $7$ cm, find the height.
9. A tank is a cylinder of radius $0.7$ m and height $2$ m, with a hemispherical bottom. Find the capacity in litres.
10. A pole consists of a cylinder of radius $7$ cm and height $50$ cm, capped by a hemisphere. Find the total surface area.
11. A vessel is in the shape of an inverted cone of height $11$ cm. Its base is closed by a flat disc of radius $2.5$ cm. Find its volume.
12. An ice-cream cone of radius $5$ and slant $13$, filled with ice cream, plus a hemispherical scoop of same radius on top. Find total volume of ice cream.

Pitfalls / Insight

(1) When computing the surface area of a combined solid, always exclude the touching surfaces. A common mistake: adding both the cylinder's top disc and the cone's bottom disc when they coincide.

(2) When computing volume, just add the basic volumes; nothing is excluded.

(3) For "cylinder length" in capsules and similar: usually the total length includes the hemispheres, so the cylinder's height is total length minus $2r$ (one diameter).

(4) For "hollowed" objects (a hemisphere scooped out of a cylinder, etc.), the surface area includes the inner curved surface that is now exposed. Sketch and label carefully.

(5) Always state the formulas you're using and the value of $\pi$ at the start. This is also good board exam hygiene.