Working out side of a triangle using given 2 sides and area?

I have been given the triangle ABC.
AB=3.2cm
BC=8.4cm
The area of triangle ABC is 10cm²
There are no right angles.

Basically, I need to find the perimeter, so I need the missing side first.
Any ideas on how to do this would be greatly appreciated, thank you.

One standard formula for the area of a triangle is
Area = 1/2 a.c sinB
Here you are given the area, a and c, i.e. the area, AB and BC
So you have: 10 = 3.2 x 8.4 x sinB
sinB = 0.3720
You now have 2 sides and the included angle.
Use cosine formula to get side b.
b^2 = a^2 + c^2 – 2a c cosB
cosB = 0.9287 (convert from from sinB)
b^2 = 8.4^2 + 3.2^2 – 2 x 8.4 x 3.2 x 0.9287
b^2 = 70.56 + 10.24 – 49.93
b^2 = 30.87
b = 5.55
So the perimeter = sum of the 3 sides = 3.2 + 8.4 + 5.55 = 17.15
I am not claiming this is the shortest method.

This entry was posted on Sunday, March 28th, 2010 at 7:01 pm and is filed under triangle area. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to “Working out side of a triangle using given 2 sides and area?”

CDA on March 29th, 2010 at 12:35 am

One standard formula for the area of a triangle is
Area = 1/2 a.c sinB
Here you are given the area, a and c, i.e. the area, AB and BC
So you have: 10 = 3.2 x 8.4 x sinB
sinB = 0.3720
You now have 2 sides and the included angle.
Use cosine formula to get side b.
b^2 = a^2 + c^2 – 2a c cosB
cosB = 0.9287 (convert from from sinB)
b^2 = 8.4^2 + 3.2^2 – 2 x 8.4 x 3.2 x 0.9287
b^2 = 70.56 + 10.24 – 49.93
b^2 = 30.87
b = 5.55
So the perimeter = sum of the 3 sides = 3.2 + 8.4 + 5.55 = 17.15
I am not claiming this is the shortest method.
References :

Area List