A triangle has sides A, B, and C. The angle between sides A and B is pi/6 and the angle between sides B and C is pi/12. If side B has a length of 3, what is the area of the triangle?

1 Answer
SagarStudy
Jan 8, 2016

Area=0.8235 square units.

Explanation:

First of all let me denote the sides with small letters a, b and c.
Let me name the angle between side a and b by /_ C, angle between side b and c by /_ A and angle between side c and a by /_ B.

Note:- the sign /_ is read as "angle".
We are given with /_C and /_A. We can calculate /_B by using the fact that the sum of any triangles' interior angels is pi radian.
implies /_A+/_B+/_C=pi
implies pi/12+/_B+(pi)/6=pi
implies/_B=pi-(pi/6+pi/12)=pi-(3pi)/12=pi-pi/4=(3pi)/4
implies /_B=(3pi)/4

It is given that side b=3.

Using Law of Sines
(Sin/_B)/b=(sin/_C)/c
implies (Sin((3pi)/4))/3=sin((pi)/6)/c

implies (1/sqrt2)/3=(1/2)/c

implies sqrt2/6=1/(2c)

implies c=6/(2sqrt2)

implies c=3/sqrt2

Therefore, side c=3/sqrt2

Area is also given by
Area=1/2bcSin/_A

implies Area=1/2*3*3/sqrt2Sin((pi)/12)=9/(2sqrt2)*0.2588=0.8235 square units
implies Area=0.8235 square units

Related questions
See all questions in Area of a Triangle
Impact of this question
1857 views around the world
You can reuse this answer
Creative Commons License