AFCAT Numerical Ability · Surface Area and Volume

Combined and Inscribed Solids

Comparing a cylinder, cone, sphere and hemisphere that share a radius or a height, the largest solid that fits inside another, and the surface or volume of solids joined together or hollowed out.

All Surface Area and Volume notes

Why this matters

These questions put two solids from the earlier pages side by side, and each one comes down to the radius or height they share. The marks are lost on surfaces: counting the face hidden where two solids meet, leaving out the inside of a cavity, or putting a floor under a tent.

Concept 1 of 3: Solids on the same base, or with equal volumes

A cone holds a third of the cylinder on the same base and height. When two solids share a radius or a height, write both volumes, cancel

\pi

and the shared measure, and what is left is the answer.

Definition

Same base and same height: cylinder $\pi r^2 h$ , cone $\tfrac13 \pi r^2 h$ , so the cylinder holds three cones.
A cylinder, a hemisphere and a cone all of radius $r$ and height $r$ hold $\pi r^3$ , $\tfrac23 \pi r^3$ and $\tfrac13 \pi r^3$ : in the ratio $3 : 2 : 1$ .
A cone standing in a hemisphere on the same flat face has height $r$ , so it holds half the hemisphere.
Equal volumes: set the two formulas equal and solve for the unknown length. Equal base and equal volume make the cone three times as tall as the cylinder.
A ratio of radii is squared in every volume; a ratio of heights is not.

Same base, same height

V_{\text{cone}} = \tfrac13 V_{\text{cylinder}},\quad V_{\text{cylinder}} : V_{\text{hemisphere}} : V_{\text{cone}} = 3 : 2 : 1 \;(h = r)

Worked example

A cone's base radius is three times a sphere's radius

r

, and the two hold the same volume. Find the cone's height in terms of

r

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Surface Area and Volume · Combined and Inscribed Solids

A cone and a cylinder are of the same height. Their radii of the bases are in ratio of 2 : 1. The ratio of their volumes is

Square the ratio of radii

A cone with three times a cylinder's radius and the same height holds

\tfrac13 \times 9 = 3

times as much, not as much. The radius enters squared.

Equal volumes do not mean equal surfaces

A cone and a cylinder on the same base with equal volumes have heights

3h

and

h

. For curved surfaces you still need the cone's slant height,

\sqrt{r^2 + 9h^2}

Drill 6 more on solids on the same base, or with equal volumes

Concept 2 of 3: The largest solid inside another

The biggest sphere that fits in a box touches its faces, so its diameter is the box's smallest measure. Draw the side view in your head and read off the radius; the wood or metal left over is the outer volume less the inner one.

Definition

Largest sphere in a cube of edge $a$ : diameter $a$ , radius $\tfrac a2$ . The smallest cube round a sphere has edge $2r$ .
Largest sphere in a cylinder: radius is the smaller of $r$ and $\tfrac h2$ .
Largest cone in a cube of edge $a$ : radius $\tfrac a2$ , height $a$ . Largest cone in a cylinder: the same base and height.
Left over (wasted) = outer volume − inner volume.

Carving one solid out of another

V_{\text{left}} = V_{\text{outer}} - V_{\text{inner}}

Worked example

The largest possible sphere is carved from a cube of edge 10 cm. Find the volume of the sphere and of the wood left, in terms of

\pi

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Surface Area and Volume · Combined and Inscribed Solids

From a solid cube of edge 3 m, a solid of largest sphere is carved out. What is the volume of solid left?

The edge is the diameter, not the radius

A sphere in a cube of edge 10 cm has radius 5 cm. Taking the radius as 10 makes the sphere eight times too big, and bigger than the cube.

In a short cylinder the height decides

A cylinder of radius 5 and height 8 holds a sphere of radius 4, not 5. Check both the width and the height.

Drill 4 more on the largest solid inside another

Concept 3 of 3: Solids joined together or hollowed out

Glue two solids together and the faces where they meet disappear inside. Drill a cavity and you lose one flat face but gain the cavity's inner surface. Volumes simply add or subtract.

Definition

Volume of a joined solid: add the parts. A cavity: subtract it.
Surface: add only what you can see. Where a cone sits on a hemisphere, or a cone on a cylinder, the joining circle is hidden on both sides.
A tent (cylinder with a cone on top) uses canvas for the two curved surfaces only: no floor, no hidden circle.
A conical cavity drilled into a cylinder on the same base leaves the curved wall, one flat end and the cone's inner surface $\pi rl$ .
Turning a rectangle about one side makes a cylinder; a right triangle turned about a leg makes a cone.

Two solids joined on a face

V = V_1 + V_2,\quad S = S_1 + S_2 - 2 \times (\text{area of the joined face})

Worked example

A tent is a cylinder of radius 12 m and height 2 m with a cone of height 9 m on top. Find the canvas needed, in terms of

\pi

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Surface Area and Volume · Combined and Inscribed Solids

A toy is in the form of a cone mounted on a hemisphere such that the diameter of the base of the cone is equal to that of the hemisphere. If the diameter of the base of the cone is 6 cm and its height is 4 cm, what is the surface area in

\text{cm}^2

of the toy? (Take

\pi = 3.14

)

No floor under a tent

Canvas covers the cylinder's wall and the cone's curved surface. Adding the base circle, or the circle where the cone meets the cylinder, overcounts.

A cavity adds a surface

Drilling a cone out of a cylinder removes one flat end but opens up the cone's inner surface. Both changes must be made.

Drill 6 more on solids joined together or hollowed out

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Solids on the same base, or with equal volumes
Same base, same height
$V_{\text{cone}} = \tfrac13 V_{\text{cylinder}},\quad V_{\text{cylinder}} : V_{\text{hemisphere}} : V_{\text{cone}} = 3 : 2 : 1 \;(h = r)$
The largest solid inside another
Carving one solid out of another
$V_{\text{left}} = V_{\text{outer}} - V_{\text{inner}}$
Solids joined together or hollowed out
Two solids joined on a face
$V = V_1 + V_2,\quad S = S_1 + S_2 - 2 \times (\text{area of the joined face})$

Watch out for (6)

Square the ratio of radii
→ Solids on the same base, or with equal volumes
Equal volumes do not mean equal surfaces
→ Solids on the same base, or with equal volumes
The edge is the diameter, not the radius
→ The largest solid inside another
In a short cylinder the height decides
→ The largest solid inside another
No floor under a tent
→ Solids joined together or hollowed out
A cavity adds a surface
→ Solids joined together or hollowed out

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