Using Trigonometry to find area of a triangle

• The area of a triangle is one half the product of the lengths of two sides and sine of the included angle.

• Area of ABC = ½ bc(sin A)

B

C

ac

bA

c

Abc sin21 Area

b a a

C

B A Substituting for h in (1)

height base Area 2

1

hc21 Area - - - - - (1)

In ,ΔACNb

hAsin

Draw the perpendicular, h, from C to BA.

hAb sin

Area of a Triangle

c21 Area Ab sin

ABC is a non-right angled triangle.

Any side can be used as the base, so

Area of a Triangle

• The formula always uses 2 sides and the angle formed by those sides

Bca sin21Cab sin

21 Abc sin

21Area = = =

Any side can be used as the base, so

Area of a Triangle

• The formula always uses 2 sides and the angle formed by those sides

c

b a a

C

B A

Bca sin21Cab sin

21 Abc sin

21Area = = =

Any side can be used as the base, so

Area of a Triangle

• The formula always uses 2 sides and the angle formed by those sides

c

b a a

C

B A

Area = = = Bca sin21Cab sin

21 Abc sin

21

Any side can be used as the base, so

Area of a Triangle

• The formula always uses 2 sides and the angle formed by those sides

c

b a a

C

B A

Area = = = Bca sin21Cab sin

21 Abc sin

21

1. Find the area of the triangle PQR.

Example

7 cm

8 cm

R

Q P

80

36 64

Solution: We must use the angle formed by the 2 sides with the given lengths.

1. Find the area of the triangle PQR.

Example

7 cm

8 cm

R

Q P

80

36 64

Solution: We must use the angle formed by the 2 sides with the given lengths.

We know PQ and RQ so use angle Q

1. Find the area of the triangle PQR.

Example

7 cm

8 cm

R

Q P

80

36 64

Solution: We must use the angle formed by the 2 sides with the given lengths.

We know PQ and RQ so use angle Q

64sin)8()7(21 Area

225 cm2 (3 s.f.)