Solving SSS Triangles
"SSS" means "Side, Side, Side"
When you know three sides of the triangle, and want to find the missing angles.
To solve an SSS triangle:
In this triangle we know the three sides:
- a = 8,
- b = 6 and
- c = 7.
Use The Law of Cosines first to find one of the angles. It doesn't matter which one. Let's find angle A first:
cos A = (b2 + c2 - a2)/2bc
A = cos-1(0.25)
Next we will find another side. We use The Law of Cosines again, this time for angle B:
cos B = (c2 + a2 - b2)/2ca
So B = 46.5674...°
Finally, we can find angle C by using 'angles of a triangle add to 180°':
So C = 180° - 75.5224...° - 46.5674...°
Now we have completely solved the triangle i.e. we have found all its angles.
This is also an SSS triangle.
In this triangle we know the three sides x = 5.1, y = 7.9 and z = 3.5. Use The Law of Cosines to find angle X first:
cos X = (y2 + z2 - x2)/2yz
= ((7.9)2 + (3.5)2 - (5.1)2)/(2×7.9×3.5)
= (62.41 + 12.25 - 26.01)/55.3
= 48.65/55.3 = 0.8797...
So X = cos-1(0.8797...)
Next we will use The Law of Cosines again to find angle Y:
cosY = (z2 + x2 - y2)/2zx
= ((3.5)2 + (5.1)2 - (7.9)2)/(2×3.5×5.1)
= (12.25 + 26.01 - 62.41)/35.7
= -24.15/35.7 = -0.6764...
So Y = cos-1(-0.6764...)
Finally, we can find angle Z by using 'angles of a triangle add to 180°':
So Z = 180° - 28.3881...° - 132.5684...°
= 19.0° correct to one decimal place.
Here is another (slightly faster) way to solve an SSS triangle:
Why do we try to find the largest angle first? That way the other two angles must be acute (less than 90°) and the Law of Sines will give correct answers.
You see, the Law of Sines is difficult to use with angles above 90°. There can be two answers either side of 90° (example: 95° and 85°), but your calculator will only give you the smaller one.
So by calculating the largest angle first using the Law of Cosines, the remaining angles will be less than 90° and the Law of Sines can be used on either of them without difficulty.
B is the largest angle, so find B first using the Law of Cosines:
Use the Law of Sines, sinC/c = sinB/b, to find angle A:
Find angle A using "angles of a triangle add to 180":
Therefore A = 47.7°, B = 104.1°, and C = 28.2°