“Teach A Level Maths”
Vol. 1: AS Core Modules
34: A Trig Formula for the
Area of a Triangle
© Christine Crisp
Trigonometry
Module C2
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with
permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
Trigonometry
3 Trig Ratios: A reminder
In a right angled triangle, the 3 trig ratios for an
angle x are defined as follows:
hypotenuse
x
sin x 
opposite
hypotenuse
opposite
Trigonometry
3 Trig Ratios: A reminder
In a right angled triangle, the 3 trig ratios for an
angle x are defined as follows:
hypotenuse
x
adjacent
cos x 
adjacent
hypotenuse
Trigonometry
3 Trig Ratios: A reminder
In a right angled triangle, the 3 trig ratios for an
angle x are defined as follows:
opposite
x
adjacent
tan x 
opposite
adjacent
Trigonometry
3 Trig Ratios: A reminder
Using the trig ratios we can find unknown angles and
sides of a right angled triangle, provided that, as
well as the right angle, we know the following:
either 1 side and 1 angle
or 2 sides
Trigonometry
3 Trig Ratios: A reminder
7
sin 30 
y

e.g. 1

y
7
30
e.g. 2


10
8
x
y 
7
sin 30
y  14
Tip: Always start with the
trig ratio, whether or not
you know the angle.
10
tan x 
8

x  51 3  (3 s.f.)
Trigonometry
Scalene Triangles
We will now find a formula for the area of a
triangle that is not right angled, using 2 sides and
1 angle.
Trigonometry
Area of a Triangle
ABC is a non-right angled triangle.
a, b and c are the sides opposite angles A, B and C
respectively. ( This is a conventional
C
way of labelling a triangle ).
b
A
a
c
B
Trigonometry
Area of a Triangle
ABC is a non-right angled triangle.
Draw the perpendicular, h, from C to BA.
C
1
Area 
base  height
2
 Area  1 c  h
2
b
a
h
A
c
N
B
Trigonometry
Area of a Triangle
ABC is a non-right angled triangle.
Draw the perpendicular, h, from C to BA.
C
1
Area 
base  height
2
 Area  1 c  h - - - - - (1)
2
b
In ΔACN ,
a
h
A
c
N
B
Trigonometry
Area of a Triangle
ABC is a non-right angled triangle.
Draw the perpendicular, h, from C to BA.
C
1
Area 
base  height
2
 Area  1 c  h - - - - - (1)
2
h
b
In ΔACN , sin A 
b
A
a
h
c
N
B
Trigonometry
Area of a Triangle
ABC is a non-right angled triangle.
Draw the perpendicular, h, from C to BA.
C
1
Area 
base  height
2
 Area  1 c  h - - - - - (1)
2
h
b
In ΔACN , sin A 
b
 b sin A  h
A
a
h
c
N
B
Trigonometry
Area of a Triangle
ABC is a non-right angled triangle.
Draw the perpendicular, h, from C to BA.
1
Area 
base  height
2
 Area  1 c  h - - - - - (1)
2
h
b
In ΔACN , sin A 
b
 b sin A  h
Substituting for h in (1) A
 Area  1 c  b sin A
2
C
a
h
c
N
B
Trigonometry
Area of a Triangle
ABC is a non-right angled triangle.
Draw the perpendicular, h, from C to BA.
1
Area 
base  height
2
 Area  1 c  h - - - - - (1)
2
h
b
In ΔACN , sin A 
b
 b sin A  h
Substituting for h in (1) A
c
1
 Area  c  b sin A
2
Area  12 bc sin A
C
a
B
Trigonometry
Area of a Triangle
Any side can be used as the base, so
Area = 1 ab sin C =
2
•
1 bc sin A =
2
The formula always uses 2
sides and the angle formed
by those sides
1 ca sin B
2
Trigonometry
Area of a Triangle
Any side can be used as the base, so
Area = 1 ab sin C =
2
•
1 bc sin A =
2
The formula always uses 2
sides and the angle formed
by those sides
A
1 ca sin B
2
C
b
a
c
B
Trigonometry
Area of a Triangle
Any side can be used as the base, so
Area = 1 ab sin C =
2
•
1 bc sin A =
2
The formula always uses 2
sides and the angle formed
by those sides
A
1 ca sin B
2
C
b
a
c
B
Trigonometry
Area of a Triangle
Any side can be used as the base, so
Area = 1 ab sin C =
2
•
1 bc sin A =
2
The formula always uses 2
sides and the angle formed
by those sides
A
1 ca sin B
2
C
b
a
c
B
Trigonometry
Example
1. Find the area of the triangle PQR.
R
80 
64 
36 
P
8 cm
7 cm
Q
Solution: We must use
the angle formed by the
2 sides with the given
lengths.
Trigonometry
Example
1. Find the area of the triangle PQR.
R
80 
36 
P
8 cm
Solution: We must use
the angle formed by the
2 sides with the given
lengths.
64 
Q
7 cm
We know PQ and RQ so use angle Q
Trigonometry
Example
1. Find the area of the triangle PQR.
R
80 
64 
36 
P
8 cm
Solution: We must use
the angle formed by the
2 sides with the given
lengths.
Q
7 cm
We know PQ and RQ so use angle Q

Area 
1 (7) ( 8) sin 64
2
 25  2 cm2
(3 s.f.)
Trigonometry
Area of a Triangle
A useful application of this formula occurs when we
have a triangle formed by 2 radii and a chord of a
circle.
A
r
C
Area  21 a b sin C
 Area 

r
Area 
B
1 r r sin 
2
1 r2
2
sin 
Trigonometry
SUMMARY
 The area of triangle ABC is given by
1
2
ab sin C or 12 bc sin A or
1 ca sin B
2
 The area of a triangle formed by 2 radii of
length r of a circle and the chord joining them
is given by
1 r2
2
where
sin
 is the angle between the radii.
Trigonometry
Exercises
1. Find the areas of the triangles shown in the
diagrams.
Y
(b)
(a) C
36 
9 cm
A
O
12 cm
28 
B
X
radius = 4 cm.,angle XOY  122
Ans: (a) 48 5 cm2 (3 s.f.)
(b) 6 78 cm2 (3 s.f.)
Trigonometry
Trigonometry
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
Trigonometry
Area of a Triangle
Any side can be used as the base, so
Area = 1 ab sin C =
2
•
1 bc sin A =
2
The formula always uses 2
sides and the angle formed
by those sides
A
1 ca sin B
2
C
b
a
c
B
Trigonometry
e.g. Find the area of the triangle PQR.
R
80 
64 
36 
P
8 cm
Solution: We must use
the angle formed by the
2 sides with the given
lengths.
Q
7 cm
We know PQ and RQ so use angle Q

Area 
1 (7) ( 8) sin 64
2
 25  2 cm2
(3 s.f.)
Trigonometry
SUMMARY
 The area of triangle ABC is given by
1
2
ab sin C or 12 bc sin A or
1 ca sin B
2
 The area of a triangle formed by 2 radii of
length r of a circle and the chord joining them
is given by
1 r2
2
where
sin
 is the angle between the radii.