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34: A Trig Formula for the34: A Trig Formula for the
Area of a TriangleArea of a Triangle
Christine Crisp
Teach A Level MathsTeach A Level Maths
Vol !: A" Core Mo#ulesVol !: A" Core Mo#ules
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Trigonometr$
Mo#ule C%
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Trigonometr$
&n a right angle#triangle' the 3 trig ratios for anangle xare #efine# as follo(s:
h$potenuse
oppositexsin
3 Trig )atios: A remin#er
oppositeh$potenuse
x
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Trigonometr$
&n a right angle#triangle' the 3 trig ratios for anangle xare #efine# as follo(s:
h$potenuse
a#*acentxcos
h$potenuse
x
a#*acent
3 Trig )atios: A remin#er
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Trigonometr$
&n a right angle#triangle' the 3 trig ratios for anangle xare #efine# as follo(s:
a#*acent
oppositextan
opposite
x
a#*acent
3 Trig )atios: A remin#er
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Trigonometr$
+sing the trig ratios (e can fin# un,no(n angles an#si#es of a right angle# triangle' provi#e# that' as(ell as the right angle' (e ,no( the follo(ing:
either! si#e an# ! angle
or% si#es
3 Trig )atios: A remin#er
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Trigonometr$
7
30
y
eg ! y7
30sin
30sin
7y
14y
810tan x
eg %
10
8
x
351
x -3 sf.
Tip: Always start with the
trig ratio, whether or not
you know the angle.
3 Trig )atios: A remin#er
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Trigonometr$
"calene Triangles
/e (ill no( fin# a formula for the area of atriangle that is not right angle#' using % si#es an#! angle
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Trigonometr$
a' ban# care the si#es opposite angles A' Ban# Crespectivel$ - This is a conventional(a$ of la0elling a triangle .
A1C is a non2right angle# triangle
A B
C
b a
c
Area of a Triangle
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Trigonometr$
ra( the perpen#icular' h' from Cto BA
N
h
height0aseArea 2
1
hc 21Area
C
b a
cA B
Area of a Triangle
A1C is a non2right angle# triangle
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Trigonometr$
ra( the perpen#icular' h' from Cto BA
N
h
height0aseArea 2
1
hc 21Area 2 2 2 2 2 (1)
&n ,4AC5
C
b a
cA B
Area of a Triangle
A1C is a non2right angle# triangle
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Trigonometr$
ra( the perpen#icular' h' from Cto BA
N
h
height0aseArea 2
1
hc 21Area 2 2 2 2 2 (1)
&n ,4AC5b
hAsin
C
b a
cA B
Area of a Triangle
A1C is a non2right angle# triangle
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Trigonometr$
hAb sin
hb a
cc
C
NA B
height0aseArea
2
1
hc 21Area 2 2 2 2 2 (1)
&n ,4AC5b
hAsin
ra( the perpen#icular' h' from Cto BA
Area of a Triangle
A1C is a non2right angle# triangle
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Trigonometr$
hb a
c
a
c
C
N B"u0stituting for hin (1) A
height0aseArea
2
1
hc 21Area 2 2 2 2 2 (1)
&n ,4AC5b
hAsin
ra( the perpen#icular' h' from Cto BA
hAb sin
Area of a Triangle
c%!Area Ab sin
h
A1C is a non2right angle# triangle
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Trigonometr$
c
Abc sin2
1
Area
b aa
C
BA"u0stituting for hin (1)
height0aseArea
2
1
hc 21Area 2 2 2 2 2 (1)
&n ,4AC5b
hAsin
ra( the perpen#icular' h' from Cto BA
hAb sin
Area of a Triangle
c%!Area Ab sin
A1C is a non2right angle# triangle
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Trigonometr$
An$ si#e can 0e use# as the 0ase' so
Area of a Triangle
The formula al(a$s uses %si#es an# the angle forme#0$ those si#es
Bca sin21Cab sin
21 Abc sin
21Area = = =
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Trigonometr$
An$ si#e can 0e use# as the 0ase' so
Area of a Triangle
The formula al(a$s uses %si#es an# the angle forme#0$ those si#es
c
baa
C
BA
Bca sin21Cab sin
21 Abc sin
21Area = = =
T i
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Trigonometr$
An$ si#e can 0e use# as the 0ase' so
Area of a Triangle
The formula al(a$s uses %si#es an# the angle forme#0$ those si#es
c
baa
C
BA
Area = = = Bca sin21Cab sin
21 Abc sin
21
T i
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Trigonometr$
An$ si#e can 0e use# as the 0ase' so
Area of a Triangle
The formula al(a$s uses %si#es an# the angle forme#0$ those si#es
c
baa
C
BA
Area = = = Bca sin21Cab sin
21 Abc sin
21
T i t
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Trigonometr$
! Fin# the area of the triangle 67)
89ample
cm
; cm
R
QP
80
36
64
"olution: /e must usethe angle forme# 0$ the% si#es (ith the givenlengths
T i t
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Trigonometr$
! Fin# the area of the triangle 67)
89ample
cm
; cm
R
QP
80
36
64
"olution: /e must usethe angle forme# 0$ the% si#es (ith the givenlengths
/e ,no( 67 an# )7 so use angle 7
T i t
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Trigonometr$
! Fin# the area of the triangle 67)
89ample
cm
; cm
R
QP
80
36
64
"olution: /e must usethe angle forme# 0$ the% si#es (ith the givenlengths
/e ,no( 67 an# )7 so use angle 7
64sin)8()7(
%!Area
%%< cm% -3 sf.
T i t
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Trigonometr$
A useful application of this formula occurs (hen (ehave a triangle forme# 0$ % ra#ii an# a chor# of acircle
Area of a Triangle
r
B
A
C
rCba sin%!Area
sinrr%!Area
sin2
r%!Area
T i t
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Trigonometr$
The area of triangle A1C is given 0$
"+MMA)=
sin2r
%
!
The area of a triangle forme# 0$ % ra#ii oflength rof a circle an# the chor# *oining themis given 0$
(here is the angle 0et(een the ra#ii
Abc sin21Cab sin
21 Bca sin
21or or
T i n m t
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Trigonometr$
! Fin# the areas of the triangles sho(n in the#iagrams
89ercises
ra#ius = 4cm'
122XOYangle
-a. -0.
X!%cm
>cm
BA
28
C
36
Y
O
-a. cm%-3 sf. -0. cm%-3 sf.548 786
Ans:
Trigonometr$
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Trigonometr$
Trigonometr$
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Trigonometr$
The follo(ing sli#es contain repeats ofinformation on earlier sli#es' sho(n (ithoutcolour' so that the$ can 0e printe# an#
photocopie#
For most purposes the sli#es can 0e printe#as ?an#outs (ith up to 6sli#es per sheet
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Trigonometr$
An$ si#e can 0e use# as the 0ase' so
Area of a Triangle
The formula al(a$s uses %si#es an# the angle forme#0$ those si#es
c
baa
C
BA
Area = = = Bca sin21Cab sin
21 Abc sin
21
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Trigonometr$
eg Fin# the area of the triangle 67)
cm
; cm
R
QP
80
36
64
"olution: /e must usethe angle forme# 0$ the% si#es (ith the givenlengths
/e ,no( 67 an# )7 so use angle 7
64sin)8()7(
%!Area
%%< cm% -3 sf.
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Trigonometr$
The area of triangle A1C is given 0$
"+MMA)=
sin2
r
%
!
The area of a triangle forme# 0$ % ra#ii oflength rof a circle an# the chor# *oining themis given 0$
(here is the angle 0et(een the ra#ii
Abc sin21Cab sin
21 Bca sin
21or or