Geometry
1
Lines and planes: distances and
angles
The angle between the edge and the base
The angle between the triangular face and the square
base
9000153601 (level 1):
Give a verbal description to the angle shown in the picture.
V
The angle between two opposite edges
The angle between the edge on a face and the base edge
9000153603 (level 1):
Give a verbal description to the angle shown in the picture.
V
ϕ
D
C
ϕ
A
B
D
C
The angle between the triangular face and the base
The angle between the edge and the base
A
B
The angle between two triangular faces
The angle between two opposite triangular faces
The angle between the edge of a triangular face and the
base edge
The angle between a triangular face and the base
The angle between two edges in the same triangular face
9000153602 (level 1):
Give a verbal description to the angle shown in the picture.
V
The angle between two triangular faces having a common
edge
9000153604 (level 1):
Give a verbal description to the angle shown in the picture.
D
A
ϕ
C
B
1
V
9000153606 (level 1):
Give a verbal description to the angle shown in the picture.
V
ϕ
D
C
A
D
B
C
ϕ
The angle between two edges in a common triangular face
A
B
The angle between two opposite triangular faces
The angle between two opposite edges
The angle between the edge on triangular face and the
base edge from the same face
The angle between two triangular faces having common
edge
The angle between the triangular face and a base edge
not in this face
The angle between two triangular faces having a common
edge
9000153605 (level 1):
Give a verbal description to the angle shown in the picture.
V
The angle between a triangular face and square base
ϕ
9000045709 (level 2):
Let ω be the angle between the solid diagonal of a box and the
base of this box. Find the expression which allows to find ω.
D
A
C
ω
B
√
The angle between opposite edges
2
tg ω =
2
The angle between a triangular face and an edge from
the opposite triangular face
√
2
cos ω =
2
√
2
sin ω =
2
cotg ω =
√
2
2
9000046407 (level 2):
Find the angle between the solid diagonal and the side of a
cube. Round the result to two decimal places.
The angle between two opposite triangular faces
The angle between two triangular faces having a common
edge
2
Identify a valid relation involving the angle α defined as an
angle between a solid diagonal and a face diagonal through
the same vertex in a cube.
a
35.26◦
45◦
54.76◦
9000046408 (level 2):
Consider a cone of base radius r and a special shape: the
shape is such that the volume of the cone is related to the
base radius by the formula V = πr3 . Find the angle between
the side of the cone and the base. Round your answer to two
decimal places.
45◦
71.57◦
a
α
a
√
√
2
tg α =
2
3
sin α =
2
cos α =
√
5
3
cotg α =
√
3
63.43◦
α = 45◦
9000046409 (level 2):
The base of a pyramid is a square with one side 2 cm. The
height of the pyramid is 4 cm. Find the angle between the side
of the pyramid and the base. Round your answer to two
decimal places.
75.96◦
70.52◦
9000120304 (level 2):
The side of a regular hexagonal prism
ABCDEF A0 B 0 C 0 D0 E 0 F 0 is a = 3 cm and the height
v = 8 cm. Find the length of the diagonal AD0 .
D0
0
E
C0
79.98◦
9000120302 (level 2):
The √
rectangular box has the sides a = 5 cm, b = 8 cm and
c = 111 cm. Find the length of the solid diagonal u.
F0
A
B0
0
v
D
E
c=
√
111
C
F
u
A
10 cm
√
73 cm
√
82 cm
a
B
√
2 8 cm
√
2 6 cm
b=8
a=5
√
10 2 cm
√
222 cm
20 cm
√
2 10 cm
9000120305 (level 2):
The side of a regular hexagonal prism
ABCDEF A0 B 0 C 0 D0 E 0 F 0 is a = 3 cm and the height
v = 8 cm. Find the angle between the diagonal AD0 and the
base plane ABC (round your result).
√
5 7 cm
9000120303 (level 2):
3
E
D0
0
H
G
C0
E
F0
A
B0
0
F
v
D
E
C
F
a
A
D
C
B
A
53
◦
◦
37
45
◦
61
◦
◦
72
54.74◦
9000120309 (level 2):
The sides of a rectangular box are a = 3 cm, b = 4 cm and
c = 12 cm. The solid diagonal is ut and the longest face
diagonal is us . Find the ratio ut : us .
B
60◦
35.26◦
39.23◦
9000121002 (level 2):
In the cube ABCDEF GH find the angle between the lines
CD and BH.
H
G
E
F
D
C
A
√
13 10 : 40
13 :
√
153
13 : 12
54.74◦
√
4 10 : 5
√
4 10 : 13
B
60◦
35.26◦
39.23◦
9000121003 (level 2):
In the cube ABCDEF GH find the angle between the lines
BSDH and AE, where the point SDH is the center of the edge
DH.
9000121001 (level 2):
In the cube ABCDEF GH find the angle between the lines
AB and AG.
4
H
G
E
H
F
G
E
F
SDH
D
C
D
C
SBC
A
70.53◦
B
29.47◦
35.26◦
A
54.74◦
63.43◦
9000121004 (level 2):
In the cube ABCDEF GH find the angle between the lines
SAE SHC and SHC SBF , where SAE , SHC and SBF are the
centers of the segments AE, HC and BF , respectively.
H
G
B
26.57◦
53.13◦
9000121006 (level 2):
In the cube ABCDEF GH find the angle between the lines
BG and GD.
H
G
E
E
36.87◦
F
F
SHC
SAE
SBF
D
D
C
C
A
A
B
60◦
53.13◦
B
26.57◦
60◦
45◦
36.87◦
53.13◦
36.87◦
9000121007 (level 2):
In the cube ABCDEF GH find the angle between the lines
SBE SAH and HC, where SBE and SAH are centers of the
segments BE and AH, respectively.
9000121005 (level 2):
In the cube ABCDEF GH find the angle between the lines
ASBC and AD, where SBC is the center of the edge BC.
5
H
G
E
H
F
G
E
F
SHD
SAH
SF C
SBE
D
C
A
60◦
D
B
26.57◦
45◦
A
53.13◦
26.57◦
9000121008 (level 2):
In the cube ABCDEF GH find the angle between the lines
DB and AG.
H
G
E
C
B
45◦
54.74◦
60◦
9000121010 (level 2):
In the cube ABCDEF GH find the angle between the lines
ESCG and AG, where SCG is the center of the CG.
H
G
F
E
F
SCG
D
C
A
90◦
D
B
45◦
35.26◦
C
A
53.13◦
54.74◦
9000121009 (level 2):
In the cube ABCDEF GH find the angle between the lines
SHD SF C and AB, where the points SHD and SF C are centers
of HD and F C, respectively.
B
19.47◦
35.26◦
60◦
9000153701 (level 2):
The picture shows a square pyramid. The side of a base
square is a = 4 cm and the height of the pyramid is v = 6 cm.
Find the angle ϕ.
6
√
2 10
.
tg ϕ =
=⇒ ϕ = 72◦ 270
2
V
9000153703 (level 2):
The picture shows a square pyramid. The side of a base
square is a = 4 cm and the height of the pyramid is v = 6 cm.
Find the angle ϕ.
V
D
C
ϕ
ϕ
A
B
6
.
=⇒ ϕ = 71◦ 340
2
6
.
tg ϕ = √ =⇒ ϕ = 64◦ 460
2 2
ϕ
2
.
tg =
=⇒ ϕ = 36◦ 520
2
6
ϕ
2
.
tg = √
=⇒ ϕ = 35◦ 60
2
2 10
tg ϕ =
D
C
A
9000153702 (level 2):
The picture shows a square pyramid. The side of a base
square is a = 4 cm and the height of the pyramid is v = 6 cm.
Find the angle ϕ.
V
ϕ
2
.
=
=⇒ ϕ = 36◦ 520
2
6
6
.
tg ϕ = √ =⇒ ϕ = 64◦ 460
2 2
ϕ
2
.
= √
=⇒ ϕ = 35◦ 60
2
2 10
√
ϕ
2 2
.
tg =
=⇒ ϕ = 50◦ 290
2
6
tg
tg
B
9000153704 (level 2):
The picture shows a square pyramid. The side of a base
square is a = 4 cm and the height of the pyramid is v = 6 cm.
Find the angle ϕ.
V
D
ϕ
C
ϕ
A
B
6
.
tg ϕ = √ =⇒ ϕ = 64◦ 460
2 2
D
C
6
.
tg ϕ =
=⇒ ϕ = 71◦ 340
2
A
ϕ
2
.
tg =
=⇒ ϕ = 36◦ 520
2
6
7
B
tg
ϕ
2
.
= √
=⇒ ϕ = 35◦ 60
2
2 10
tg
ϕ
2
.
=
=⇒ ϕ = 36◦ 520
2
6
V
6
.
tg ϕ = √ =⇒ ϕ = 64◦ 460
2 2
tg
√
ϕ
2 2
.
=
=⇒ ϕ = 50◦ 290
2
6
9000153705 (level 2):
The picture shows a square pyramid. The side of a base
square is a = 4 cm and the height of the pyramid is v = 6 cm.
Find the angle ϕ.
V
D
C
ϕ
A
ϕ
tg ϕ =
B
√
2 10
.
=⇒ ϕ = 72◦ 270
2
6
.
tg ϕ = √ =⇒ ϕ = 64◦ 460
2 2
D
C
tg ϕ =
A
√
2 2
ϕ
.
=⇒ ϕ = 50◦ 290
tg =
2
6
tg
2
ϕ
.
=
=⇒ ϕ = 36◦ 520
2
6
√
ϕ
2 2
.
tg =
=⇒ ϕ = 50◦ 290
2
6
B
6
.
tg ϕ = √ =⇒ ϕ = 64◦ 460
2 2
tg
6
.
=⇒ ϕ = 71◦ 340
2
9000128801 (level 3):
The base ABCD of a square pyramid ABCDV has side 6 cm.
The height of the pyramid is 4 cm. The point M is the middle
of the side CV . Find the distance between the point M and
the plane ABC.
V
ϕ
2
.
=⇒ ϕ = 35◦ 60
= √
2
2 10
9000153706 (level 2):
The picture shows a square pyramid. The side of a base
square is a = 4 cm and the height of the pyramid is v = 6 cm.
Find the angle ϕ.
M
D
C
A
2 cm
B
√
34
cm
2
5
cm
2
9000128802 (level 3):
The base ABCD of a square pyramid ABCDV has side 6 cm.
The height of the pyramid is 4 cm. The point M is the middle
8
of the side CV . Find the distance between the point M and
the line BC.
V
C
A
B
√
5
cm
2
34
cm
2
√
7
cm
2
D
43.31◦
C
A
59.04◦
45◦
M
B
√
97
cm
2
B
9000128806 (level 3):
The base ABCD of a square pyramid ABCDV has side 6 cm.
The height of the pyramid is 4 cm. The point M is the middle
of the side CV . Find the angle between the line AM and the
plane ABC. Round to two decimal places.
V
M
D
C
A
9000128803 (level 3):
The base ABCD of a square pyramid ABCDV has side 6 cm.
The height of the pyramid is 4 cm. The point M is the middle
of the side CV . Find the distance between the point M and
the line AD.
V
√
6 cm
9000128805 (level 3):
The base ABCD of a square pyramid ABCDV has side 6 cm.
The height of the pyramid is 4 cm. Find the angle between the
line BV and the plane ABC. Round to two decimal places.
V
M
D
√
15 34
cm
5
24
cm
5
106
cm
2
√
D
65
cm
2
C
A
9000128804 (level 3):
The base ABCD of a square pyramid ABCDV has side 6 cm.
The height of the pyramid is 4 cm. Find the distance between
the line AD and the plane BCV .
V
D
A
17.45◦
B
34.50◦
18.32◦
9000128807 (level 3):
The base ABCD of a square pyramid ABCDV has side 6 cm.
The height of the pyramid is 4 cm. Find the angle between the
planes DCV and ABC. Round to two decimal places.
C
B
9
V
The lengths of a side, face diagonal and solid diagonal through
the vertex A in a rectangular box ABCDEF GH are
|AB| = 6 cm, |AC| = 10 cm, |AG| = 15 cm. Find the surface
area.
H
G
D
C
E
A
53.13◦
F
B
59.04◦
D
43.31◦
9000128808 (level 3):
The base ABCD of a square pyramid ABCDV has side 6 cm.
The height of the pyramid is 4 cm. Find the angle between the
planes ADV and BCV . Round to two decimal places.
V
C
A
B
√ 96 + 140 5 cm2
600 cm2
√
236 5 cm2
√ 48 + 70 5 cm2
√
240 5 cm2
D
C
A
73.74◦
2
9000120307 (level 2):
The lengths of a side, base diagonal and solid diagonal
through the vertex A in a rectangular box ABCDEF GH are
|AB| = 6 cm, |AC| = 10 cm, |AG| = 15 cm. Find the volume of
the box.
H
G
B
36.87◦
61.93◦
Volume and surface area formulas
E
F
9000120301 (level 2):
√
The length of a solid diagonal in a cube is u = 2 6 cm. Find
the surface area.
D
C
A
√
240 5 cm3
√
u=2 6
2
48 cm
2
24 cm
√
B
900 cm3
√
300 5 cm3
√
600 2 cm3
√
240 2 cm3
2
24 2 cm
√
16 2 cm2
√
9000120308 (level 2):
The height v of a regular hexagonal prism
√ is a double of its
side a. The volume of the prism is 648 3 cm3 . Use this
information to find the length of the longest solid diagonal in
the prism.
2
12 6 cm
9000120306 (level 2):
10
It is a congruent mapping (preserves shapes and dimensions).
It is a similar mapping (preserves shapes, may change
dimensions and position).
v
It is an identity mapping (objects are mapped to itself).
9000149302 (level 1):
Given a line in a plane, find how many points are mapped
onto itself with reflection through this line.
a
√
12 2 cm
√
10 6 cm
√
12 6 cm
√
6 10 cm
√
432 cm
infinitely many
none
one
two
9000120310 (level 2):
The base of a rectangular box ABCDEF GH has sides
|AB| = 6 cm and |BC| = 8 cm. The angle between the solid
diagonal AG and the base ABC is 60◦ . Find the volume of the 9000149303 (level 1):
box.
Which of the following answers contains three letters with
H
G
point symmetry? (A letter has a point of symmetry if there
exists a point, such that the reflection through this point
maps the letter into itself.)
E
F
O, N, Z
D
A, M, O
A, O, B
H, O, U
9000149304 (level 1):
How many points of symmetry there exist for a rhombus? (A
point is a point of symmetry of the rhombus if the reflection
through this point maps the rhombus into itself.)
A rhombus (opposite sides are parallel and all sides have equal
length).
C
8cm
6cm
A
√
480 3 cm3
960 cm3
√
288 3 cm3
B
√
160 3 cm3
240 cm3
3
one
Symmetry and geometric
transformations
none
two
four
9000149305 (level 1):
Given a translation T of a plane, find the lines which are
mapped to the same line by T .
All lines parallel to the translation vector are mapped
into itself.
9000149301 (level 1):
Which of the following is true for a reflection through a line?
All lines perpendicular to the translation vector are mapped into itself.
11
There are no lines which are mapped into itself by the
translation.
9000149310 (level 1):
The dilatation defined by a center and a scale factor k = −1 is
equivalent to another geometric mapping. Which one?
Every line is mapped into itself by the translation.
9000149306 (level 1):
Given a translation of a plane, find the property of a line
obtained by translating a line r. The line r is neither parallel
not perpendicular to the translation vector.
The resulting line is parallel to the line r
The resulting line is perpendicular to the translation vector.
The resulting line is perpendicular to the line r
The resulting line is the line r. (The line r is mapped into
itself.)
9000149307 (level 1):
The rotation through an angle α = 180◦ is equivalent to some
other geometric mapping. Which one?
reflection through the point
reflection through the line
translation
9000149308 (level 1):
Consider a rotation through an angle either α = 180◦ or
α = 360◦ . How many lines which are mapped into itself exists
for this rotation?
infinitely many
none
one
two
9000149309 (level 1):
Consider a dilatation which maps A onto B. The center is the
dilatation is S. Find a correct statement.
The point S is on the line through the points A and B.
The points S, A and B form a right triangle ABS.
The distance from S to A is smaller than the distance
from A to B.
The points S, A and B form a triangle ABS with at least
two sides of equal length.
12
reflection through the point
translation
reflection through the line
rotation