4.10 3D shapes and volume

Learning objectives

4.10A Names of solidsRecognise and name cubes, cuboids, prisms, pyramids, cylinders, spheres and cones.4.10B Faces, edges and verticesunderstand the terms ‘face’, ‘edge’ and ‘vertex’ in the context of 3D solids 4.10C The surface area of simple shapes using the areafind the surface area of simple shapes using the area formulae for triangles and rectangles 4.10D The surface area of a cylinderfind the surface area of a cylinder 4.10E The volume of prismsfind the volume of prisms, including cuboids and cylinders, using an appropriate formula 4.10F Between units of volume within the metric systemconvert between metric units of volume, including cm³ and m³, and use 1 litre = 1000 cm³ 4.10HA The surface area and volume of a sphere and a rightfind the surface area and volume of a sphere and a right circular cone using relevant formulae

Name a solid from the structure of its surfaces and cross-sections, not from the way a perspective sketch happens to look.

Solid	Defining feature
cube / cuboid	six square / rectangular faces
prism	identical parallel end faces and constant cross-section
pyramid	one polygonal base; triangular faces meet at one vertex
cylinder	two parallel circular ends and one curved surface
sphere	every surface point is the same distance from the centre
cone	circular base and curved surface meeting at one vertex

Identify any repeated parallel cross-section, then check the number and shape of plane faces and whether a curved surface or single apex is present.

A cylinder is a circular prism in some broad usage, but the syllabus expects the specific name 'cylinder'; a pyramid narrows to a point while a prism does not.

A face is a flat surface of a polyhedron, an edge is where two faces meet, and a vertex is a corner where edges meet. Curved solids may also be described using curved surfaces and circular boundaries.

Solid	Faces / surfaces	Edges	Vertices
cube or cuboid	6	12	8
triangular prism	5	9	6
square-based pyramid	5	8	5
cylinder	2 plane faces + 1 curved surface	2 circular boundaries	0
cone	1 plane face + 1 curved surface	1 circular boundary	1

For a prism with an $n$ -sided end face: faces $=n+2$ , edges $=3n$ , vertices $=2n$ . Trace systematically so hidden dashed edges are included.

Do not count a drawn diagonal, construction line or curved outline twice; perspective drawings can hide genuine edges but do not create new ones.

Total surface area is the sum of the areas of every exposed face. A net or face inventory turns the 3D solid into separate triangles and rectangles that can be checked.

Step	Action
1	identify the congruent end faces
2	list every lateral rectangle with its two dimensions
3	calculate each face area and group equal faces
4	omit only faces explicitly open, joined or unpainted

For a right prism, lateral area equals perimeter of cross-section $\times$ prism length; then add the two end areas when both are exposed.

Volume units are cubic, but surface area units are square. A hidden face still contributes unless it is joined internally or the question excludes it.

Unrolling a cylinder gives a rectangle of width $2\pi r$ and height $h$ . Its curved area is $2\pi rh$ , so total surface area is $2\pi rh+2\pi r^2$ .

Cylinder	Surface area
closed	$2\pi rh+2\pi r^2$
open at one end	$2\pi rh+\pi r^2$
curved surface only	$2\pi rh$

Confirm whether the given circular measure is radius or diameter, find any missing height from other data if needed, and include exactly the exposed circular ends.

The circle formula $\pi r^2$ is used for each end; $2\pi r$ is a length and becomes an area only after multiplication by height.

Every prism has volume $V=(\text{cross-sectional area})\times(\text{perpendicular length})$ . Thus a cuboid has $V=lwh$ and a cylinder has $V=\pi r^2h$ .

Step	Decision
1	identify the constant end cross-section
2	calculate its area in square units
3	multiply by the perpendicular prism length
4	convert units before comparing capacity, cost or count

For packing, volume alone gives an upper bound; whole boxes must also fit by their dimensions. For filling, divide the required volume by a rate or container capacity and round according to context.

Do not multiply by a sloping edge unless it is the perpendicular distance through which the cross-section is repeated.

A linear conversion factor is cubed for volume. Since $1\text{ m}=100\text{ cm}$ , $1\text{ m}^3=100^3\text{ cm}^3=1{,}000{,}000\text{ cm}^3$ .

Relationship	Equivalent volume
$1\text{ m}^3$	$1{,}000{,}000\text{ cm}^3$
$1\text{ litre}$	$1000\text{ cm}^3$
$1\text{ m}^3$	$1000\text{ litres}$

Write the one-dimensional relationship, cube its factor for cubic units, then multiply toward smaller units or divide toward larger units. Use the litre bridge only after units are compatible.

Multiplying by $100$ converts a length, not a volume; multiplying by $100^2$ converts an area, not a volume.

For a sphere, surface area is $4\pi r^2$ and volume is $\frac43\pi r^3$ . For a right circular cone, volume is $\frac13\pi r^2h$ , curved area is $\pi rl$ , and total area is $\pi rl+\pi r^2$ .

Shape feature	Relationship
right cone	$l^2=r^2+h^2$
hemisphere volume	$\frac23\pi r^3$
hemisphere curved area	$2\pi r^2$
solid hemisphere total area	$3\pi r^2$ including its base

For joined solids, add volumes but count only external surfaces. For a hollow or removed part, subtract its volume or exposed area. Similar cones scale lengths by $k$ , areas by $k^2$ and volumes by $k^3$ .

Cone surface area uses slant height $l$ , while cone volume uses perpendicular height $h$ . A joined circular face is internal and must not be counted in external surface area.

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4.10 3D shapes and volume

Learning objectives

Recognise and name common solids

Identify faces, edges and vertices

Find surface areas from faces and nets

Find the surface area of a cylinder

Find volumes of prisms and cylinders

Convert metric volumes and litres

Find sphere and right-cone measures

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