AREA OF TRIANGLE USING SINE FORMULA

Area of the triangle is a half of product of two sides and the side included angle.

Consider the triangle given below, in which the sides opposite angles A, B and C are labelled a, b and c respectively.

In the triangle given above

AB  =  c (base) and

CN  =  h(height)

In triangle ANC,

sin A  =  Opposite side / hypotenuse

sin A  =  h/b

h  =  b (sin A)

So, area of triangle ABC  =  (1/2) ⋅ base ⋅ height

=  (1/2) ⋅ c ⋅ (b sin A)

Area of triangle  =  (1/2) ⋅ (bc sin A)

Find the area of the triangle given below :

Example 1 :

Solution :

<B  =  45, a  =  12 cm and c  =  13 cm.

Area of triangle  =  (1/2) ⋅ (ac sin B)

=  (1/2) ⋅ (13) ⋅ (12) sin 45

=  (1/2) ⋅ (13) ⋅ (12) (0.707)

=  55.146 cm²

Example 2 :

Solution :

<C  =  82, a  =  28 km and c  =  25 km

Area of triangle  =  (1/2) ⋅ (ac sin C)

=  (1/2) ⋅ (28) ⋅ (25) sin 82

=  (1/2) ⋅ (28) ⋅ (25) (0.990)

=  346.5 km²

So, the area of the given triangle is 347 km².

Example 3 :

Solution :

<A  =  112, c  =  6.4 cm and b  =  7.8 cm

Area of triangle  =  (1/2) ⋅ (bc sin A)

=  (1/2) ⋅ (7.8) ⋅ (6.4) sin 112

=  (1/2) ⋅ (7.8) ⋅ (6.4) (0.927)

=  23.13 cm²

So, the area of the given triangle is 23.13 cm².

Example 4 :

Solution :

<A  =  84, b  =  32 m and c  =  27 m

Area of triangle  =  (1/2) ⋅ (bc sin A)

=  (1/2) ⋅ (32) ⋅ (27) sin 84

=  (1/2) ⋅ (32) ⋅ (27) (0.994)

=  429.40 m²

So, the area of the given triangle is 430 cm².

Example 5 :

Solution :

<A  =  125, b  =  12.2 cm and c  =  10.6 cm

Area of triangle  =  (1/2) ⋅ (bc sin A)

=  (1/2) ⋅ (12.2) ⋅ (10.6) sin 125

=  (1/2) ⋅ (12.2) ⋅ (10.6) (0.819)

=  52.95 cm²

So, the area of the given triangle is 53cm².

Example 6 :

Find the area of a parallelogram with sides 6.4 cm and 8.7 cm and one interior angle 64^o.

Solution :

In a parallelogram, opposite angles are equal and opposite sides are equal.

So, by drawing the diagonal we can divide the parallelogram into two triangles of equal area.

<A  =  64, b  =  6.4 cm and c  =  8.7 cm

Area of triangle  =  (1/2) ⋅ (bc sin A)

=  (1/2) ⋅ (6.4) (8.7) sin 64

=   (1/2) ⋅ (6.4) (8.7) (0.898)

=  25

Area of parallelogram  =  2(25)

=  50 cm²

Problem 7 :

If triangle ABC has area 150 cm², find the value of x.

Solution :

<B  =  75, a  =  x cm and c  =  14 cm

Area of triangle  =  (1/2) ⋅ (ac sin B)

=  (1/2) ⋅ (x) (14) sin 75

=   7x (0.965)

=  6.755x

6.755x  =  150

x  =  150/6.755

x  =  22.2 cm

So, the value of x is 22.2 cm.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.