CDS Mathematics · Mensuration 3D

Solids Inside Solids

One solid fitted inside another so that it just touches: a cube in a sphere, a cylinder round a sphere, a sphere in a cone, and balls or cones touching each other.

All Mensuration 3D notes CDS Mathematics strategy

Why this matters

The hardest page of the chapter: more than half its questions are HARD. The method is always the same — find the ONE length the two solids share (a diagonal, a diameter, a height) — and then the question is the formulas of the earlier pages.

Concept 1 of 3: Cube, cylinder and sphere nested

A cube inside a sphere touches it at its corners, so the cube's space diagonal is the sphere's diameter. A cylinder or sphere inside a cube touches its faces, so its diameter is the cube's edge.

Definition

Cube in a sphere: $a\sqrt3 = 2R$ .
Sphere in a cube: $2r = a$ .
Cylinder just enclosing a sphere: radius $r$ , height $2r$ ; its curved surface equals the sphere's surface, $4\pi r^2$ .
Cuboid in a sphere: the sphere's diameter is the space diagonal.
Cylinder inscribed in a sphere of radius $R$ with height $2x$ : radius $\sqrt{R^2 - x^2}$ .

Cube in a sphere

a\sqrt3 = 2R

Worked example

A cube is inscribed in a sphere of radius

3\sqrt3

cm. Find the cube's volume.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2020 · CDS (I) 2020 — Elementary Mathematics · Q88Moderate

Example 1 · Mensuration 3D · Solids Inside Solids

A right circular cylinder just encloses a sphere. If

p

is the surface area of the sphere and

q

is the curved surface area of the cylinder, then which one of the following is correct ?

Drill 7 more on cube, cylinder and sphere nested

Concept 2 of 3: A sphere inside a cone

Cut through the axis. The cone becomes an isosceles triangle and a sphere touching its sides becomes the triangle's incircle. So the sphere's radius is the triangle's area divided by its semi-perimeter.

Definition

Axial section: triangle with base $2r$ , height $h$ , equal sides $l$ .
Inscribed sphere's radius $= \dfrac{\text{area}}{s} = \dfrac{rh}{r + l}$ .
A sphere touching the cone's sides has its centre $\dfrac{\rho}{\sin\alpha}$ from the apex ( $\alpha$ the semi-vertical angle).
Two spheres stacked in a cone: $\sin\alpha = \dfrac{R - r}{R + r}$ .

Sphere in a cone

\rho = \frac{rh}{r + l}

Worked example

A cone has base radius

6

cm and height

8

cm. Find the radius of the sphere that fits inside it, touching the base and the sides.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (II) 2022 — Elementary Mathematics · Q71Hard

Example 2 · Mensuration 3D · Solids Inside Solids

Consider the following for the items that follow : A conical vessel of radius 12 cm and height 16 cm is filled with water. A sphere is lowered into water and its size is such that it touches the sides of the vessel and it is just immersed.

What is the radius of the sphere ?

Drill 4 more on a sphere inside a cone

Concept 3 of 3: Balls and cones that touch

Equal balls or cones standing on a plane and touching each other have their centres (or their base centres) at the corners of an equilateral triangle. A fourth ball resting on three sits above the centre of that triangle.

Definition

Three equal circles of radius $r$ touching: centres form an equilateral triangle of side $2r$ , circumradius $\dfrac{2r}{\sqrt3}$ .
A fourth ball on three: its centre is $\sqrt{(2r)^2 - \dfrac{4r^2}{3}} = 2r\sqrt{\tfrac23}$ above the other centres, which are themselves $r$ above the ground.
The apexes of three equal cones touching on a plane lie on a circle of radius $\dfrac{2r}{\sqrt3}$ .

Fourth ball on three

\text{height of centre} = r + 2r\sqrt{\tfrac23}

Worked example

Three equal balls of radius

\sqrt3

cm lie on a table, each touching the other two. Find the radius of the circle through their centres.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2024 · CDS (II) 2024 — Elementary Mathematics · Q93Hard

Example 3 · Mensuration 3D · Solids Inside Solids

Three identical cones each with base radius 3 cm are placed on their bases so that each is touching the other two. There will be one and only circle that would pass through each of the vertices of the cones. What is the area of the circle ?

Measure from the table, not from the lower centres

The fourth ball's centre is

2r\sqrt{\tfrac23}

above the other CENTRES. The question usually asks for its height above the plane, which adds one more

r

Drill 1 more on balls and cones that touch

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)

Cube, cylinder and sphere nested
Cube in a sphere
$a\sqrt3 = 2R$
A sphere inside a cone
Sphere in a cone
$\rho = \frac{rh}{r + l}$
Balls and cones that touch
Fourth ball on three
$\text{height of centre} = r + 2r\sqrt{\tfrac23}$

Watch out for (1)

Measure from the table, not from the lower centres
→ Balls and cones that touch

Test yourself on Mensuration 3D

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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