1. The Area of Scalene Triangle
• If two sides and one angle are given
Look at the triangle ABD :
B 𝒉
𝒔𝒊𝒏 𝑨 = 𝒄
↔ 𝒉 = 𝒄 𝒔𝒊𝒏 𝑨
c a
h
A D C
b
𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡
Area of triangle = 2
𝑏 ×𝑐 sin 𝐴
=
2
1
Area = 𝑏𝑐 sin 𝐴
2

2. Now, we change the position of base and height line :
C
Look at the triangle BCD :
𝒉
b a 𝒔𝒊𝒏 𝑩 = ↔ 𝒉 = 𝒂 𝒔𝒊𝒏 𝑩
𝒂
h
A D B
c
𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡
Area of triangle = 2
𝑐 ×𝑎 sin 𝐵
=
2
1
Area = 𝑎𝑐 sin 𝐵
2

3. Look at the triangle ACD :
A
𝒉
𝒔𝒊𝒏 𝑪 = 𝒃
↔ 𝒉 = 𝒃 𝒔𝒊𝒏 𝑪
c b
h
B D C
a
𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡
Area of triangle = 2
𝑎 ×𝑏 sin 𝐶
=
2
1
Area = 𝑎𝑏 sin 𝐶
2

4. So, we get the area of scalene triangle :
1
Area = 𝑏𝑐 sin 𝐴
2
1
Area = 𝑎𝑐 sin 𝐵
2
1
Area = 𝑎𝑏 sin 𝐶
2

5. Example
B
A C
If the angle of A = 30°, the angle of C = 45°, the
length of c = 5 cm. Determine the area !

6. Before we find the area, we find the
undetermined elements.
The angle of B = 180° − 30° + 45° = 105°
𝑐 𝑎
=
sin 𝐶 sin 𝐴
5 𝑎
=
sin 45° sin 30°
5 𝑎
1 = 1
2
2 2
5 5
a= = 2
2 2

7. 1
The area = 𝑐. 𝑎. sin 105°
2
1 5
= 5 2 0,97
2 2
25
= 2 0,97
4
2
= 8,57 𝑐𝑚

8. • If all of sides are given
By the rules of cosine (𝑎2 = 𝑏 2 + 𝑐 2 −
2𝑏𝑐 cos 𝐴), we can get the area of scalene
triangle.
2𝑏𝑐 cos 𝐴 = 𝑏 2 + 𝑐 2 − 𝑎2
𝑏 2 + 𝑐 2 − 𝑎2
cos 𝐴 =
2𝑏𝑐

9. In the formula of trigonometry, 𝑠𝑖𝑛2 𝐴 + 𝑐𝑜𝑠 2 𝐴 = 1
𝑠𝑖𝑛2 𝐴 = 1 − 𝑐𝑜𝑠 2 𝐴
2
𝑏2 +𝑐 2 −𝑎2
=1 −
2𝑏𝑐
𝑏2 +𝑐 2 −𝑎2 𝑏2 +𝑐 2 −𝑎2
= 1+ 1−
2𝑏𝑐 2𝑏𝑐
2𝑏𝑐+𝑏2 +𝑐 2 −𝑎2 2𝑏𝑐−𝑏2 +𝑐 2 −𝑎2
=
2𝑏𝑐 2𝑏𝑐
(𝑏+𝑐)2 −𝑎2 𝑎2 −(𝑏−𝑐)2
=
2𝑏𝑐 2𝑏𝑐
(𝑏+𝑐+𝑎)(𝑏+𝑐−𝑎)(𝑎+𝑏−𝑐)(𝑎−𝑏+𝑐)
=
(2𝑏𝑐)2

10. 1
Because s = × circumference of triangle
2
1
s= 𝑎+ 𝑏+ 𝑐
2
2𝑠 × 2 𝑠 − 𝑎 × 2(𝑠 − 𝑏) × 2(𝑠 − 𝑐)
𝑠𝑖𝑛2 𝐴 =
(2𝑏𝑐)2
16𝑠(𝑠−𝑎)(𝑠−𝑏)(𝑠−𝑐)
=
4(𝑏𝑐)2
4𝑠(𝑠−𝑎)(𝑠−𝑏)(𝑠−𝑐)
=
(𝑏𝑐)2
2
sin 𝐴 = 𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
𝑏𝑐

11. 1
The area = 𝑏𝑐 sin 𝐴
2
1 2
= 𝑏𝑐 𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
2 𝑏𝑐
The area = 𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)

12. Example
Determine the area of triangle ABC if AB= 10 cm,
BC= 8 cm, and AC= 6 cm !
Answer :
1
s= 8 + 6 + 10 = 12 𝑐𝑚
2
The area = 𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
= 12(12 − 8)(12 − 6)(12 − 10)
= 576 = 24 𝑐𝑚2