CDS Mathematics · Mensuration 3D

Spheres, Hemispheres & Shells

Volume and surface of a sphere and a hemisphere, hollow shells and their mass, and the circle in which a plane cuts a sphere.

All Mensuration 3D notes CDS Mathematics strategy

Why this matters

A small page with two formulas to learn and two ideas to use: a shell is the outer sphere minus the inner one, and a plane cutting a sphere gives a right triangle between the centre, the cut and the rim.

Concept 1 of 3: Sphere and hemisphere

A sphere's surface is four times the area of its great circle; its volume is a third of the radius times the surface. A hemisphere has half the volume, but its total surface adds the flat circle.

Definition

Sphere: $V = \dfrac43\pi r^3$ , $S = 4\pi r^2$ .
Hemisphere: $V = \dfrac23\pi r^3$ ; curved surface $2\pi r^2$ ; total $3\pi r^2$ .
Numerically equal surface and volume: $4\pi r^2 = \dfrac43\pi r^3$ gives $r = 3$ .
A sphere cut into $n$ equal wedges by planes through one diameter: each wedge has $\dfrac1n$ of the curved surface plus two flat semicircles.

Sphere and hemisphere

V_{\text{sphere}} = \tfrac43\pi r^3, \quad S = 4\pi r^2, \quad V_{\text{hemi}} = \tfrac23\pi r^3

Worked example

Find the total surface area of a solid hemisphere of radius

7

cm.

(\pi = \tfrac{22}{7})

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q67Moderate

Example 1 · Mensuration 3D · Spheres, Hemispheres and Shells

The volume of a hemisphere is 155232

\text{cm}^3

. What is the radius of the hemisphere ?

Drill 4 more on sphere and hemisphere

Concept 2 of 3: Hollow shells and their mass

A shell is the outer sphere with the inner sphere removed. Its metal is

\dfrac43\pi(R^3 - r^3)

; multiply by the density for the mass.

Definition

Shell volume $= \dfrac43\pi(R^3 - r^3)$ , with $R$ the outer and $r$ the inner radius.
Mass $=$ volume $\times$ density. In g/cm $^3$ , the volume must be in cm $^3$ .
Given diameters, halve them first.

Spherical shell

V = \tfrac43\pi(R^3 - r^3)

Worked example

A hollow shell has inner radius

4

cm and outer radius

5

cm, made of metal of density

3

g/cm

^3

. Find its mass.

(\pi = \tfrac{22}{7})

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (II) 2021 — Elementary Mathematics · Q68Moderate

Example 2 · Mensuration 3D · Spheres, Hemispheres and Shells

A hollow spherical shell is made of a metal of density 7 g/cm

^3

. If its internal and external radii are 3 cm and 6 cm respectively, then what is the mass of the shell ? (take

\pi = \frac{22}{7}

)

Drill 1 more on hollow shells and their mass

Concept 3 of 3: A plane cutting a sphere

A flat cut through a sphere is a circle. Its radius, its distance from the centre and the sphere's radius form a right triangle. Bowls, pots and water levels in a spherical vessel all use this one triangle.

Definition

A plane at distance $d$ from the centre of a sphere of radius $R$ cuts a circle of radius $\rho = \sqrt{R^2 - d^2}$ .
A bowl filled to depth $t$ : the water surface is $R - t$ from the centre.
A pot of height $H$ cut from a sphere of radius $R$ (with $H > R$ ): the cut is $H - R$ above the centre.

Circle of a cut

\rho^2 + d^2 = R^2

Worked example

A spherical bowl of radius

13

cm holds water to a depth of

8

cm. Find the radius of the water surface.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (I) 2025 — Elementary Mathematics · Q59Moderate

Example 3 · Mensuration 3D · Spheres, Hemispheres and Shells

A pot is made from a hollow sphere of inner radius 20 cm by cutting its upper portion horizontally. The height of the pot is 30 cm.

What is the inner radius of the circular opening of the pot so formed ?

Drill 2 more on a plane cutting a sphere

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)

Sphere and hemisphere
Sphere and hemisphere
$V_{\text{sphere}} = \tfrac43\pi r^3, \quad S = 4\pi r^2, \quad V_{\text{hemi}} = \tfrac23\pi r^3$
Hollow shells and their mass
Spherical shell
$V = \tfrac43\pi(R^3 - r^3)$
A plane cutting a sphere
Circle of a cut
$\rho^2 + d^2 = R^2$

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.

Take the Mensuration 3D chapter test Drill all 10 questions