Geometry of 3-D Objects: Faces, Vertices, Edges & Nets — Worksheet & Memo

∑ DANEMATHICS

FREE CAPS MATHS RESOURCES

Grade 9 · Geometry of 3-D Objects

Geometry of 3-D Objects: Faces, Vertices, Edges & Nets

MARKING GUIDELINE

Marks

Duration

55 minutes

Questions

Name:

Class:

Date:

Mark

/ 34

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
Euler's formula for a polyhedron is …
(1)
A)F + V + E = 2
B)F + V − E = 2✓
C)F − V + E = 2
D)F × V ÷ E = 2
Answer: B — Faces plus vertices minus edges equals 2.
- A — for a cube that gives 26, not 2
- C — for a cube that gives 10
- D — for a cube that gives 4
1.2
A tetrahedron has 4 faces and 4 vertices. Its number of EDGES is …
(1)
A)8
B)6✓
C)4
D)12
Answer: B — 4 + 4 − E = 2 gives E = 6.
- A — added instead of using the formula
- C — assumed edges equal faces
- D — that is a cube
1.3
A prism with a hexagonal cross-section has … faces.
(1)
A)6
B)8✓
C)12
D)7
Answer: B — Six rectangles plus the two hexagonal ends.
- A — counted only the rectangles
- C — that is the number of vertices
- D — that is a hexagonal PYRAMID
1.4
The difference between a prism and a pyramid is that a pyramid has …
(1)
A)two identical parallel bases
B)ONE base and triangular faces meeting at an apex✓
C)only curved surfaces
D)no vertices
Answer: B — A prism has two identical parallel ends; a pyramid tapers to a point.
- A — that describes a PRISM
- C — a pyramid has flat faces
- D — a pyramid has an apex, which is a vertex
1.5
Which solid has NO vertices at all?
(1)
A)cube
B)sphere✓
C)cone
D)square pyramid
Answer: B — A sphere is a single smooth curved surface.
- A — a cube has 8
- C — a cone has one, at its apex
- D — a square pyramid has 5
1.6
A net of a cube consists of …
(1)
A)4 squares
B)6 squares✓
C)8 squares
D)6 rectangles
Answer: B — One for each face.
- A — that would leave two faces open
- C — a cube has only 6 faces
- D — the faces of a cube are squares
1.7
Euler's formula does NOT apply to a cylinder because a cylinder …
(1)
A)has too many faces
B)has a CURVED surface✓
C)has no edges
D)is too large
Answer: B — Euler's formula applies to polyhedra, whose faces are all flat.
- A — it has very few faces
- C — it does have circular edges
- D — size is irrelevant
1.8
An octahedron has 8 faces and 12 edges. Its number of VERTICES is …
(1)
A)4
B)6✓
C)8
D)20
Answer: B — 8 + V − 12 = 2 gives V = 6.
- A — that is a tetrahedron
- C — confused vertices with faces
- D — that is an icosahedron
1.9
Which one of the following is NOT a Platonic solid?
(1)
A)cube
B)square pyramid✓
C)tetrahedron
D)octahedron
Answer: B — A Platonic solid has identical regular faces; a square pyramid mixes a square with triangles.
- A — six identical squares
- C — four identical triangles
- D — eight identical triangles
1.10
A cube has 6 faces, 8 vertices and 12 edges. Euler's formula gives …
(1)
A)26
B)2✓
C)10
D)14
Answer: B — 6 + 8 − 12 = 2.
- A — added all three
- C — subtracted in the wrong order
- D — used 6 + 8

Answer grid — marking guideline

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[12 MARKS]

The solid below is a cube.

2.1
Write down the number of faces, vertices and edges.
(3)
F = 6 (1), V = 8 (1), E = 12 (1)
2.2
Verify Euler's formula for this solid, and hence determine how many edges a solid with 5 faces and 6 vertices must have.
(3)
F + V − E = 6 + 8 − 12 = 2 ✓ (2)
5 + 6 − E = 2 gives E = 9 edges (1)
2.3
A different solid has 5 faces and 6 vertices. Use Euler's formula to find its number of edges, and name the solid.
(3)
5 + 6 − E = 2 (1)
E = 9 (1)
It is a triangular prism (1)
2.4
Explain why Euler's formula cannot be applied to a cylinder, and name the part of a cylinder that prevents it.
(3)
A cylinder has a CURVED surface (1)
Euler's formula applies only to polyhedra — solids whose faces are all flat, with straight edges and vertices (1)
It is the curved side (the lateral surface) that disqualifies it (1)

Question 3

[12 MARKS]

Answer the questions below about 3-D objects. F is the number of faces, V the number of vertices and E the number of edges.

3.1
A tetrahedron has 4 faces and 4 vertices. Use Euler's formula to determine its number of edges.
(3)
F + V − E = 2 (1)
4 + 4 − E = 2 (1)
E = 6 (1)
3.2
An octahedron has 8 faces, 6 vertices and 12 edges. Verify Euler's formula for it, and hence determine how many edges meet at each vertex.
(3)
8 + 6 − 12 = 2 ✓ (2)
Each edge has 2 ends, so 12 × 2 ÷ 6 = 4 edges meet at every vertex (1)
3.3
A prism has a hexagonal cross-section. Determine its number of faces, vertices and edges.
(4)
F = 6 rectangles + 2 hexagons = 8 (1)
V = 6 + 6 = 12 (1)
E = 6 + 6 + 6 = 18 (1)
Check: 8 + 12 − 18 = 2 ✓ (1)
3.4
Explain the difference between a prism and a pyramid.
(2)
A prism has two identical parallel faces joined by rectangles (1); a pyramid has ONE base and all its other faces are triangles meeting at an apex (1)

TOTAL: 34 marks

This question paper consists of 3 questions.