Learn8300Papers 1–3 · Higher tierGeometry and measuresPyramids, cones and spheres

Geometry and measures · Subtopic 9 of 17

Pyramids, Cones and Spheres

A pyramid or cone holds exactly one third of the prism with the same base and height. A sphere has volume $\tfrac43\pi r^3$ and surface area $4\pi r^2$ . Composite solids are built from these pieces: add the volumes, and count only the outside surfaces.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Find the volume of pyramids and cones using $\tfrac13\times$ base $\times$ height.
Find the slant height and curved surface area of a cone.
Find the volume and surface area of spheres and hemispheres.
Work with frustums and composite solids.

Pause and think

A cone and a cylinder have the same circular base and the same height. How many cones of water would fill the cylinder?

Section 2 of 14

Core idea

Start hereThree cones fill a cylinder

Fill a cone with water and pour it into a cylinder of the same base and height. It takes exactly three cones. Pointed solids hold one-third of their matching prism.

The big idea

Pyramid/cone: $V=\frac13\times\text{base}\times h$ ; cone curved surface $\pi rl$ . Sphere: $V=\frac43\pi r^3$ , $SA=4\pi r^2$ .

1. The one-third rule

A cone fits inside a cylinder three times over

l

(slant height),

r

and

h

form a right-angled triangle:

l^2=r^2+h^2

V_{\text{cone}}=\tfrac13\pi r^2h,\qquad\text{curved }SA=\pi rl,\qquad l=\sqrt{r^2+h^2}

$h$: perpendicular height.
$l$: slant height.

Worked exampleCone $r=3$ , $h=4$

Slant height, volume, total surface.

Goal: Values.
Strategy: Pythagoras for $l$ .

$l=\sqrt{9+16}=5$
Right triangle $r,h,l$ .
$V=\tfrac13\pi(9)(4)=12\pi=37.7$
Volume.
$\pi(3)(5)+\pi(9)=24\pi=75.4$
Curved + base.

Answer

5

;

37.7

cm³;

75.4

cm².

✓ CheckCylinder of same size:

36\pi

; one-third

12\pi

✓.

2. Spheres

Worked exampleSphere $r=6$

Volume, surface; solid hemisphere.

Goal: Values.
Strategy: Formulas; hemisphere adds a flat circle.

$\tfrac43\pi(216)=288\pi=905$
Volume.
$4\pi(36)=144\pi=452$
Surface.
$V=144\pi;\ SA=72\pi+36\pi=108\pi$
Half, plus the flat face $\pi r^2$ .

Answer

905

cm³;

452

cm²; hemisphere

144\pi

108\pi

✓ CheckHemisphere surface isn't half the sphere: it gains a circle ✓.

Worked exampleMelting and recasting

Sphere radius 6 melted into cones $r=3$ , $h=4$ .

Goal: Number of cones.
Strategy: Volume is conserved.

$288\pi\div12\pi$
Divide volumes.
$=24$
Evaluate.

Answer

24

cones.

✓ CheckWhole number: no waste ✓.

Common misconceptionSlant vs vertical height Volume uses

h

; curved surface uses

l

Pause and think

Square pyramid base 6, height 10. Volume?

Summary

Pointed solids: $\tfrac13$ .
Cone $\pi rl$ ; sphere $4\pi r^2$ .

Slant height: Apex to base edge along the surface.

HistoryArchimedes' favourite result. Archimedes proved that a sphere fits inside a cylinder with exactly

\tfrac23

of its volume and

\tfrac23

of its surface area. He was so proud of it that he asked for a sphere inside a cylinder to be carved on his tomb. The Roman writer Cicero reported finding the overgrown tomb about

140

years later by that carving.

Fun factWhy bubbles are round. Of all shapes with a given volume, the sphere has the smallest surface area. Soap film pulls itself as tight as possible, so a floating bubble always becomes a sphere.

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