Question

A triangle has sides A, B, and C. The angle between sides A and B is `pi/4` and the angle between sides B and C is `pi/12`. If side B has a length of 15, what is the area of the triangle?

Answer 1

The area of a triangle is given by `Area=1/2bh` where `b` is the length of the base and `h` is the perpendicular height. Draw a diagram and calculate as shown below that the area is `88.35` `cm^2`.

Explanation:

Drawing a clear diagram is essential for this kind of problem. Mine is not perfectly to scale, but it allows me to have clear in my own mind what information I have and what I'm trying to achieve.

(I've placed the image at the bottom of the page because the formatting works better that way.)

We know that the area of a triangle is given by:

`Area=1/2bh`

We already know that the base is `15` `cm`, but we don't know the perpendicular height. I have constructed a right angled triangle, so we could use trig, but we don't know how far along the `15` `cm` side the added line hits.

We could use sine rule or cosine rule in the main triangle (they work in all triangles, not just right-angled ones), but we still need a little more information.

In radians, the angles in a triangle add up to `pi` (or `180^o`, if we worked in degrees). We already have `angleAB`=`pi/4=(3pi)/12` and `angleBC`=`pi/12` for a total of `(4pi)/12`, so `angleAC` (all of it) must be `(8pi)/12`.

Now we can use the sine rule to find the length of `A` or `C` or both:

`C/sinc=B/sinc to C = Bsinc/sinb = 15(sin(pi/4))/sin((8pi)/12) = 12.2` `cm`

(for our purposes here, lowercase letters represent angles and uppercase represent sides, though the convention is usually the other way around)

Now that we know the length of `C`, we can use trig to find the height of the vertical line: the height of the triangle. Let's call that `'D'`:

`cos(pi/12)=D/C to D=Ccos(pi/12)=12.2cos(pi/12)=11.78` `cm`

Now we know the base and height of the triangle, we can calculate its area:

`Area=1/2bh=1/2*15*11.78=88.35` `cm^2`