Checkpoint 6 Maths · Geometry and Measure · 6Gg.04 · 6Gg.06

3D Shapes, Compound Shapes and Nets

Learn the faces, edges and vertices of 3D shapes, describe and sketch compound shapes, and recognise and draw nets of cubes, cuboids, prisms and pyramids.

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Identify, describe and sketch compound 3D shapes (shapes made by joining two or more 3D shapes). (6Gg.04)
Identify and sketch different nets for cubes, cuboids, prisms and pyramids. (6Gg.06)
This means: know the names and properties (faces, edges, vertices) of common 3D shapes, say which shapes a compound shape is made from, draw 3D shapes on plain or isometric dot paper, and recognise and draw the flat patterns (nets) that fold up to make them.

1. Key words

Word / phrase	Meaning / use
3D shape (solid)	مجسم — a shape with length, width and height
face	وجه — a flat (or curved) surface of a 3D shape
edge	حرف / ضلع — where two faces meet
vertex (plural: vertices)	رأس (رؤوس) — a corner where edges meet
cube / cuboid	مكعب / متوازي مستطيلات
prism	منشور — two identical end faces joined by rectangles; the same shape all along
pyramid	هرم — a base with triangular faces that meet at one point (the apex)
apex	القمة — the top point of a pyramid or cone
compound shape	مجسم مركب — a shape made by joining two or more 3D shapes
net	شبكة (مفرود المجسم) — a flat shape that folds up to make a 3D shape
isometric dot paper	ورق نقط متساوي القياس — paper with dots in a triangle pattern, used to draw 3D shapes

2. Faces, edges and vertices

A cuboid: a face is a flat surface, an edge is a line where two faces meet, a vertex is a corner.

In drawings of 3D shapes, dashed lines show edges that are hidden at the back. Count them too: they are real edges.

Common 3D shapes. Dashed lines are hidden edges.

3D shape	Faces	Edges	Vertices	Shapes of the faces
cube	6	12	8	6 squares
cuboid	6	12	8	6 rectangles (opposite faces identical)
triangular prism	5	9	6	2 triangles + 3 rectangles
square-based pyramid	5	8	5	1 square + 4 triangles
triangle-based pyramid (tetrahedron)	4	6	4	4 triangles
pentagonal prism	7	15	10	2 pentagons + 5 rectangles
hexagonal prism	8	18	12	2 hexagons + 6 rectangles
cylinder	3 (2 flat, 1 curved)	2 (curved)	0	2 circles + 1 curved surface
cone	2 (1 flat, 1 curved)	1 (curved)	1 (the apex)	1 circle + 1 curved surface

Prisms and pyramids are named after their end or base. A prism has the same shape all the way through: two identical end faces joined by rectangles. A pyramid has one base, and triangles that meet at the apex.

Patterns you can use (generalising)

A prism whose end has n sides has n rectangles, so n + 2 faces altogether. Example: a hexagonal prism has 6 + 2 = 8 faces.
A pyramid whose base has n sides has n triangles, so n + 1 faces altogether. Example: a pentagonal pyramid has 5 + 1 = 6 faces.
Check by imagining the shape: count the ends (or base) first, then the faces around the sides.

3. Compound 3D shapes

A compound 3D shape is made by joining two or more 3D shapes together. Many real objects are compound shapes: a pencil (cylinder + cone), a house (cuboid + triangular prism), a tower with a pointed roof (cube + square-based pyramid), a set of steps (cuboids).

To describe a compound shape, say: (1) which shapes it is made from, (2) how they are joined (on top, at the side, which faces touch), and (3) its properties if asked (number of faces, edges, vertices).

Shape P is a cube with a square-based pyramid on top. Shape Q is a cylinder with a cone on top.

Name the parts. Look for the shapes you know: a pointed top is often a pyramid or a cone; a round body is a cylinder.
Count each part's faces, then think about the join: the two faces that are glued together are now inside the shape, so they are no longer faces.
Look for faces that join flat. If two faces end up side by side in the same flat surface, they make one bigger face.
Check by counting on the picture, including faces at the back (look for dashed lines).

A cuboid and a triangular prism of the same length and width make a house shape. The front rectangle and the front triangle join flat to make one pentagon, so the house shape is really a pentagonal prism.

Two cuboids joined to make an L-shaped step. The dotted line shows where the two cuboids meet; it is not an edge, because the front is one flat L-shaped face.

المجسم المركب بالعربي

المجسم المركب = مجسمين أو أكتر لازقين في بعض. عشان توصفه: قول هو متكون من إيه (مثلاً مكعب وفوقه هرم)، وإزاي متلزقين. ولما تعد الأوجه: الوجهين اللي اتلزقوا في بعض بيختفوا جوه، ولو وجهين بقوا سطح واحد مستوي بنعدهم وجه واحد.

4. Sketching 3D shapes

On plain paper (the quick way for a cuboid or prism):

Draw the front face (for a cuboid a rectangle, for a triangular prism a triangle).
Draw the same face again a little up and to the right (this is the back face).
Join each corner of the front face to the matching corner of the back face with straight lines.
Make the edges you could not see from the front into dashed lines.

On isometric dot paper (the dots make triangles, not squares):

Turn the paper so the dots make vertical columns.
Vertical edges are drawn as vertical lines.
Edges going left-back and right-back are drawn along the slanting lines of dots. Never draw horizontal lines.
Draw one cube at a time: each cube shows three faces, a top and two sides (a hexagon outline).
Start with the cube nearest to you and build backwards and upwards; shade the top faces to make the drawing clear.

A compound shape made of 5 cubes, sketched on isometric dot paper. Every corner of every cube is on a dot.

5. Nets of a cube

A net is a flat 2D shape that can be cut out and folded to make a 3D shape, with no gaps and no overlaps. A net of a cube is made of 6 squares, but not every arrangement of 6 squares works. There are 11 different nets of a cube. Here are some of them, and some arrangements that do NOT work.

Imagine folding each one. The arrangements marked ✗ fail: when folded, two squares land in the same place (overlap) and a gap is left somewhere else.

Quick checks for a cube net

There must be exactly 6 squares.
A 2 by 2 block of squares can never be part of a cube net.
There can never be 5 squares in a straight row (only 4 can wrap around the cube).
Fold in your head: pick one square as the bottom and fold the others up around it. If two squares need the same place, it is not a net.

Opposite faces. When a net folds into a cube, some faces end up opposite each other. In a straight line of squares, squares that are one apart are opposite (1st and 3rd, 2nd and 4th). Opposite faces never touch in the net.

In this cross-shaped net: A is opposite F, B is opposite D, and C is opposite E.

This is how a dice is made: the numbers on opposite faces add up to 7 (1 and 6, 2 and 5, 3 and 4).

6. Nets of cuboids, prisms and pyramids

A cuboid has 3 pairs of identical rectangles. In the net, edges that join must be the same length.

A net of a cuboid 4 cm long, 3 cm wide and 2 cm high: two 4 × 3 rectangles (top, bottom), two 4 × 2 rectangles (front, back) and two 3 × 2 rectangles (sides).

A prism: draw the rectangles in a row (one for each side of the end shape), then attach one end face to the top and one to the bottom. A pyramid: draw the base, then attach one triangle to each side of the base.

Nets of a triangular prism (2 triangles + 3 rectangles), a square-based pyramid (1 square + 4 triangles) and a triangle-based pyramid (4 triangles).

Each 3D shape has more than one net. For example, the end triangles of the prism could be attached to any of the three rectangles, and on opposite ends or the same ends of different rectangles, as long as they do not overlap when folded.

Net contains…	It folds to make…
6 squares	a cube
6 rectangles (3 identical pairs)	a cuboid
2 triangles + 3 rectangles	a triangular prism
2 hexagons + 6 rectangles	a hexagonal prism
1 square + 4 triangles	a square-based pyramid
4 triangles	a triangle-based pyramid (tetrahedron)
1 pentagon + 5 triangles	a pentagonal pyramid

الشبكة (المفرود)

الشبكة هي شكل مسطح لو قصيناه وطبّقناه يعمل المجسم من غير فراغات ولا تداخل.
المكعب ليه ١١ شبكة مختلفة — كلها ٦ مربعات، بس مفيش أبداً ٥ مربعات في صف واحد ولا مربع ٢×٢.
المنشور: مستطيل لكل ضلع + وجهين للنهايتين. الهرم: القاعدة + مثلث لكل ضلع.

7. Worked examples

Worked example 1 Look at shape P in section 3 (a cube with a square-based pyramid on top). (a) Which two 3D shapes make shape P? [1] (b) How many faces, edges and vertices does shape P have? [2]

(a) A cube and a square-based pyramid.
(b) Faces: the cube has 6, but its top is covered, so 5 are left (4 sides + bottom). The pyramid has 5, but its square base is covered, so 4 triangles are left. Total 9 faces. The triangles slope, so they do not join flat with the cube's sides.
Edges: cube 12 + pyramid 8 = 20, but the 4 edges of the joining square are shared, so 20 − 4 = 16 edges.
Vertices: cube 8 + pyramid 5 = 13, but the 4 corners of the joining square are shared, so 13 − 4 = 9 vertices.
Common mistake: just adding 6 + 5 = 11 faces, forgetting the two joined faces are hidden.

Worked example 2 A triangular prism is placed exactly on top of a cuboid of the same length and width to make a house shape (see section 3). Hassan says the house shape has 6 + 5 − 2 = 9 faces. Explain why Hassan is wrong and give the correct number. [2]

When the shapes join, the front rectangle of the cuboid and the front triangle of the prism are in the same flat surface, so they make one pentagon-shaped face. The same happens at the back.
So 9 − 2 = 7 faces: 2 pentagons + 5 rectangles (2 walls, 2 roof faces, 1 floor). The house shape is a pentagonal prism.
Mark-scheme style: 1 mark for explaining the faces join flat into one face; 1 mark for 7.

Worked example 3 Look at the arrangements marked ✗ in section 5. Explain why the arrangement "5 in a row" is not a net of a cube. [1]

If you fold a row of squares around a cube, 4 squares go all the way round (front, side, back, side). The 5th square would land on top of the 1st square (overlap), so one face would be missing.
Answer: A row of 5 squares overlaps when folded, because only 4 faces fit around a cube.

Worked example 4 The dice net below needs a number on the blank square. Opposite faces add up to 7. What number goes on the blank square? [1]

Worked example 4: the dice net.

In a row of four, the 1st and 3rd squares are opposite (3 and 4: 3 + 4 = 7 ✓), and the 2nd and 4th squares are opposite.
So the blank square is opposite 2: 7 − 2 = 5.

Worked example 5 A cuboid is 4 cm long, 3 cm wide and 2 cm high. (a) How many rectangles of each size are in its net? [1] (b) Sketch a net of the cuboid. [2]

(a) 2 rectangles 4 cm × 3 cm, 2 rectangles 4 cm × 2 cm and 2 rectangles 3 cm × 2 cm. (Each pair is a face and the face opposite it.)
(b) See the net in section 6. Mark-scheme style: 1 mark for six rectangles of the correct sizes; 1 mark for a correct arrangement that folds into the cuboid (joining edges the same length).
Common mistake: drawing six identical squares or rectangles — that would be a cube, not this cuboid.

Worked example 6 A net is made of 1 pentagon and 5 triangles. (a) Name the 3D shape. [1] (b) How many edges and vertices does it have? [1]

(a) One base + triangles meeting at a point = pyramid. The base is a pentagon, so it is a pentagonal pyramid.
(b) Edges: 5 around the base + 5 going up to the apex = 10 edges. Vertices: 5 on the base + 1 apex = 6 vertices.

Common mistakes

Forgetting the hidden faces, edges or vertices at the back (dashed lines are real edges).
Adding the faces of two joined shapes without taking away the faces that are glued together.
Counting two faces that join flat as two faces (the house shape has 7 faces, not 9).
Calling a pyramid a prism (or the other way round): a prism has two identical ends; a pyramid has one base and an apex.
Thinking any 6 squares make a cube net: check for a 2 by 2 block, 5 in a row or two squares folding into the same place.
Cuboid nets with edges that do not match: the edges that meet when folded must be the same length.
Drawing horizontal lines on isometric paper: only vertical and slanting lines are used.

How Checkpoint asks this

"Here is a net. Name the 3D shape it makes." Count the faces and look at their shapes.
"Tick (✓) the nets that make a cube." Fold each one in your head; cross out ones with a 2 by 2 block or 5 in a row.
"The net is folded to make a cube. Which face is opposite face A?" Use the one-apart rule.
"Draw one more square to complete the net of a cube." There may be several correct places: any one is fine.
"Write the number of faces, edges and vertices" in a table, sometimes for a compound shape.
"Which two 3D shapes have been joined to make this shape?" Name both shapes fully (square-based pyramid, not just pyramid).
"Sketch the shape on the isometric paper" or "complete the drawing": vertical lines stay vertical, every corner on a dot.
Use a pencil and ruler for nets and sketches; they do not need to be exact sizes unless the question says so.

8. Practice

Remember

Faces are surfaces, edges are where faces meet, vertices are corners. Count the hidden ones too.
A prism has two identical ends joined by rectangles; a pyramid has a base and triangles meeting at an apex.
A compound shape is two or more shapes joined: name each part fully. Glued faces disappear; faces that join flat become one face.
A net folds into a 3D shape with no gaps or overlaps. A cube has 11 nets: never a 2 by 2 block, never 5 in a row.
In a cube net, squares one apart in a line are opposite. On a dice, opposite faces add to 7.
On isometric dot paper, vertical edges stay vertical and the other edges follow the slanting rows of dots.

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