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Name of Lecturer: Mr. J. Agius
Course: HVAC 2
Lesson 35
Chapter 4. Trigonometry
Trigonometry in 3D
To find angles or lengths in solid figures, a right-angled triangle containing the unknown
quantity has to be identified. This triangle should then be drawn separately from the solid
figure.
E
Example 1
The figure is a pyramid on a square base ABCD.
The edges of the base are 30cm long and the
height EH of the pyramid is 42cm.
Find
C
a) the length of AC
b) the angle EAH.
D
Answer
H
B
A
a)
Therefore
x2 = 900 + 900 = 1800
x = 42.42
Therefore
AC = 42.4 cm correct to 3 s.f.
b)
C
AC is diagonal of the square base ABCD.
Using ABC,  B = 90o, AB = BC = 30cm
AC2 = AB2 + BC2 (Pythagoras theorem)
x
A
B
30
 EAH is in AHE, in which  H = 90o and
AH = ½ AC = 21.21 cm
E
42 (opp)
opp
42
tan A 

 1.9802
adj 21.21
 A = 63.2o
30
A
21.21
H
(adj)
4 Trigonometry
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Name of Lecturer: Mr. J. Agius
Course: HVAC 2
The Angle Between a Line and a Plane
The angle between the line PQ and the plane ABCD is defined as the angle between PQ and
its projection on the plane ABCD.
Q
Q
B
B
C
C
A
A
P
P
N
D
D
Draw a perpendicular, QN, from Q to the plane. (N is called the foot of the perpendicular.)
Join P to N.
The required angle is QPN. (This angle is tucked under the line.)
The line PN is called the projection of the line PQ on the plane.
(Note: if the plane does not look horizontal, it may help you to see which line is the
perpendicular if you turn the page and look at the diagram from a different angle.)
The Angle Between Two Planes
P
Q
To find the angle between two planes we need to find two lines to act as the arms of the
angle.
The two lines, one in each plane, must meet on the joining line, PQ, of the two planes and
each must be at right angles to the joining line.
A
C
P
Q
One possible angle is ARC.
Notice that there are any number of pairs of lines that meet at a given point on the joining line
but only a pair at right angles to PQ gives the required angle. DEF is not a possible angle
but DQF is possible.
4 Trigonometry
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Name of Lecturer: Mr. J. Agius
Course: HVAC 2
D
F
P
E
Q
It can be helpful to use a section through a solid when trying to identify the angle between
two of its faces. The section, or cut, must be made at right angles to the edge where the two
faces meet, cutting this edge at P, say. Then the angle between the faces is the angle at Q in
the section. For example, in a right pyramid, the angle between the base and a sloping face
can be found from a section formed by a cut through the vertex perpendicular to the base.
Example 2
ABCD is a right pyramid of height 5cm. Its base ABCD is a rectangle in which AB = 6cm
and BC = 8cm.
a)
Find the angle between EA and the plane ABCD.
E
b)
Find the angle between the planes EBC and ABCD.
Answer
a)
(EF is the perpendicular from E to ABCD)
EAF is the required angle.
1
AF  AC
A
2
In ABC, AC = 10cm (3, 4, 5)
Therefore
AF = 5cm
Therefore
EAF = 45o (rt-angled isos. )
D
F
G
6cm
C
8cm
B
The angle between EA and ABCD is 45o.
b)
(BC is the joining line for the planes. Form symmetry, EG and FG are both
perpendicular to BC at G.)
EGF is the required angle.
In EFG, F = 90o.
5cm
E
opp 5
  1.6666...
adj 3
5cm (opp)
EGF = tan 1 (1.6666...)  59 o
tan G 
So
E
A
F
The angle between the planes is 59o.
F
4 Trigonometry
G
Page 3
Name of Lecturer: Mr. J. Agius
Course: HVAC 2
Exercise 1
V
1)
For this pyramid, whose height is 10m,
a)
Calculate the length of DB
b)
Calculate the length of VB
c)
Find the angle that
10m
X
10m
VB makes with the base ABCD
d)
C
10m
D
Calculate the length of VY, where
B
A
Y
Y is the midpoint of AB
e)
Draw the section containing the
angle between VY and the face ABCD.
f)
Calculate the angle described in (e)
g)
Find the angle between the planes
VAB and ABCD.
E
2)
13 cm
F
D
C
For this wedge,
A
20 cm
12 cm
a)
calculate the lengths of AF and FC
b)
draw the section containing the angle between the diagonal FC and the
ABCD
c)
find the size of the angle that FC makes with the base ABCD.
d)
find the size of the angle between the planes FBCE and ABCD.
3)
In this cuboid, AB = 7cm, AE = 6cm
B
base
H
G
and BC = 10cm.
Find the size of the angle between
a)
EC and the plane ABCD
b)
FD and the plane ABFE
c)
the planes EBCH and AEHD
d)
the planes EFCD and CDHG.
4 Trigonometry
E
F
6 cm
D
C
10 cm
A
7 cm
B
Page 4
Name of Lecturer: Mr. J. Agius
Course: HVAC 2
4)
E
D
Q
A
N
P
C
B
The great pyramid of Cephren at Gizeh in Egypt has a square base of side 215m and is 225m
high. In the diagram E is the vertex of the right pyramid and ABCD is its base. The diagonals
of the base intersect at N; P and Q are the midpoints of AB and DC respectively.
a)
A tunnel runs from the entrance P to the burial chamber at N. Find PN. How far is N
from A?
b)
How far is it
i) from P to the top
ii) from A to the top?
c)
A climber wishes to climb from the base of the pyramid to the top. Where should he
start
i) to make the shortest climb
ii) to climb in one straight line at the smallest angle to the horizontal?
5)
The diagram shows a part of a sea wall whose
constant cross-section is a right-angled triangle.
The road is at the level of the top, DC, and the
beach starts at AB. Jim clambers from the road to
the beach straight down the pat DA whereas Pete
goes along the path DB.
Find
a)
C
D
3m
B
the length of each path.
8m
4m
b)
the inclination to the horizontal of each path.
A
A
6)
The diagram shows a corner block
of a concrete edging system.
AB = BC = BD = 6cm.
Find the angle that the sloping face,
ACD, makes with the horizontal base, BCD.
D
B
C
4 Trigonometry
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