Area of a TriangleArea of a Triangle

A

B

12cm C

10cm

Example 1 : Find the area of the triangle ABC.

50o

(i) Draw in a line from B to AC

(ii) Calculate height BD

D

o BDSin50 =

10oBD = 10 Sin50 = 7.66

2

1

2

0.5 12 7.66 46

Area base height

cm

(iii) Area

7.66cm

Area of a TriangleArea of a Triangle

Q

P

20cm R

12cm

Example 2 : Find the area of the triangle PQR.

40o

(i) Draw in a line from P to QR

(ii) Calculate height PS

S

o PSSin40 =

10oPS = 12 Sin40 = 7.71

2

1

2

0.5 20 7.71 77.1

Area base height

cm

(iii) Area

7.71cm

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. Know the formula for the Know the formula for the area of any triangle.area of any triangle.

1. To explain how to use the Area formula for ANY triangle.

Area of ANY TriangleArea of ANY Triangle

2.2. Use formula to find area of Use formula to find area of any triangle given two any triangle given two length and angle in length and angle in between.between.

General Formula forGeneral Formula forArea of ANY TriangleArea of ANY Triangle

Consider the triangle below:

Ao Bo

Co

ab

c

h

Area = ½ x base x height 1

2A c h

What does the sine of Ao equal

sin o hA

b

Change the subject to h. h = b

sinAoSubstitute into the area formula

1sin

2oA c b A

1sin

2oA bc A

Area of ANY TriangleArea of ANY Triangle

A

B

C

A

aB

b

Cc

The area of ANY triangle can be found by the following formula.

sin1

Area= ab C2

sin1

Area= ac B2

sin1

Area= bc A2

Another version

Another version

Key feature

To find the areayou need to knowing

2 sides and the angle in between (SAS)

Area of ANY TriangleArea of ANY Triangle

A

B

C

A

20cmB

25cm

Cc

Example : Find the area of the triangle.

sinC1

Area= ab2

The version we use is

30o

120 25 sin30

2oArea

210 25 0.5 125Area cm

Area of ANY TriangleArea of ANY Triangle

D

E

F

10cm

8cm

Example : Find the area of the triangle.

sin1

Area= df E2

The version we use is

60o

18 10 sin 60

2oArea

240 0.866 34.64Area cm

What Goes In The Box What Goes In The Box ??

Calculate the areas of the triangles below:

(1)

23o

15cm

12.6cm

(2)

71o

5.7m

6.2m

A =36.9cm2

A =16.7m2

Key feature

Remember (SAS)

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. Know how to use the sine Know how to use the sine rule to solve REAL LIFE rule to solve REAL LIFE problems involving problems involving lengths.lengths.

1. To show how to use the sine rule to solve REAL LIFE problems involving finding the length of a side of a triangle .

Sine RuleSine Rule

C

B

A

Sine RuleSine Rule

a

b

c

The Sine Rule can be used with ANY triangle as long as we have been given enough information.

Works for any Triangle

a b c= =

SinA SinB SinC

Deriving the rule

B

C

A

b

c

a

Consider a general triangle ABC.

The Sine Rule

Draw CP perpendicular to BA

P

CPSinB CP aSinB

a

CP

also SinA CP bSinAb

aSinB bSinA

aSinBb

SinA

a bSinA SinB

This can be extended to

a b cSinA SinB SinC

or equivalentlySinA SinB SinCa b c

Calculating Sides Calculating Sides Using The Sine RuleUsing The Sine Rule

10m

34o

41o

a

Match up corresponding sides and angles:

sin 41oa

10

sin 34o

Rearrange and solve for a. 10sin 41

sin34

o

oa 10 0.656

11.740.559

a m

Example 1 : Find the length of a in this triangle.

A

B

C

sin sin sino

a b c

A B C

Calculating Sides Calculating Sides Using The Sine Using The Sine

RuleRule

10m133o

37o

d

sin133od

10

sin 37o

10sin133

sin 37

o

od

10 0.731

0.602d

=

12.14m

Match up corresponding sides and angles:

Rearrange and solve for d.

Example 2 : Find the length of d in this triangle.

C

D

E

sin sin sino

c d e

C D E

What goes in the Box What goes in the Box ??

Find the unknown side in each of the triangles below:

(1) 12cm

72o

32o

a

(2)

93o

b47o

16mm

a = 6.7cm b =

21.8mm

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. Know how to use the sine Know how to use the sine rule to solve problems rule to solve problems involving angles.involving angles.

1. To show how to use the sine rule to solve problems involving finding an angle of a triangle .

Sine RuleSine Rule

Calculating Angles Calculating Angles

Using The Sine Using The Sine RuleRule

Example 1 :

Find the angle Ao

A

45m

23o

38m

Match up corresponding sides and angles:

45

sin oA 38

sin 23o

Rearrange and solve for sin Ao

45sin 23sin

38

ooA = 0.463 Use sin-1 0.463 to find Ao

1sin 0.463 27.6o oA

sin sin sin

a b c

A B C

B

C

Calculating Angles Calculating Angles

Using The Sine Using The Sine RuleRule

143o

75m

38m

X

38

sin oX

75

sin143o

38sin143sin

75

ooX = 0.305

1sin 0.305 17.8o oX

Example 2 :

Find the angle Xo

Match up corresponding sides and angles:

Rearrange and solve for sin Xo

Use sin-1 0.305 to find Xo

Y

Z

sin sin sin

x y z

X Y Z

What Goes In The Box What Goes In The Box ??

Calculate the unknown angle in the following:

(1)

14.5m

8.9m

Ao

100o (2)

14.7cm

Bo

14o

12.9cm

Ao = 37.2o

Bo = 16o