Solving SSS Triangles

"SSS" means "Side, Side, Side"

SSS Triangle

"SSS" is when we know three sides of the triangle, and want to find the missing angles.

To solve an SSS triangle:

We use the "angle" version of the Law of Cosines:

cos(C) = a2 + b2 − c2 2ab

cos(A) = b2 + c2 − a2 2bc

cos(B) = c2 + a2 − b2 2ca

(they are all the same formula, just different labels)

Example 1

trig SSS example 6,7,8

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
cos(A) = (62 + 72 − 82) / (2×6×7)
cos(A) = (36 + 49 − 64) / 84
cos(A) = 0.25
A = cos-1(0.25)
A = 75.5224...°
A = 75.5° to one decimal place.

Next we find another side. We use the Law of Cosines again, this time for angle B:

cos(B) = (c2 + a2 − b2)/2ca
cos(B) = (72 + 82 − 62)/(2×7×8)
cos(B) = (49 + 64 − 36) / 112
cos(B) = 0.6875
B = cos-1(0.6875)
B = 46.5674...°
B = 46.6° to one decimal place

Finally, we can find angle C by using "angles of a triangle add to 180°":

C = 180° − 75.5224...° − 46.5674...°
C = 57.9° to one decimal place

Now we have completely solved the triangle i.e. we have found all its angles.

The triangle can have letters other than ABC:

Example 2

trig SSS example

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
cos(X) = ((7.9)2 + (3.5)2 − (5.1)2)/(2×7.9×3.5)
cos(X) = (62.41 + 12.25 − 26.01)/55.3
cos(X) = 48.65/55.3 = 0.8797...
X = cos-1(0.8797...)
X = 28.3881...°
X = 28.4° to one decimal place

Next we will use The Law of Cosines again to find angle Y:

cos(Y) = (z2 + x2 − y2)/2zx
cos(Y) = −24.15/35.7 = −0.6764...
cos(Y) = (12.25 + 26.01 − 62.41)/35.7
cos(Y) = −24.15/35.7 = −0.6764...
Y = cos-1(−0.6764...)
Y = 132.5684...°
Y = 132.6° to one decimal place.

Finally, we can find angle Z by using "angles of a triangle add to 180°":

Z = 180° − 28.3881...° − 132.5684...°
Z = 19.0° to one decimal place

Another Method

Here is another (slightly faster) way to solve an SSS triangle:

Largest Angle?

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.

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 a calculator will only give you the smaller one.

So by calculating the largest angle first using the Law of Cosines, the other angles are less than 90° and the Law of Sines can be used on either of them without difficulty.

Example 3

trig SSS example

B is the largest angle, so find B first using the Law of Cosines:

cos(B) = (a2 + c2 − b2) / 2ac
cos(B) = (11.62 + 7.42 − 15.22) / (2×11.6×7.4)
cos(B) = (134.56 + 54.76 − 231.04) / 171.68
cos(B) = −41.72 / 171.68
cos(B) = −0.2430...
B = 104.1° to one decimal place

Use the Law of Sines, sinC/c = sinB/b, to find angle A:

sin(C) / 7.4 = sin(104.1°) / 15.2
sin(C) = 7.4 × sin(104.1°) / 15.2
sin(C) = 0.4722...
C = 28.2° to one decimal place

Find angle A using "angles of a triangle add to 180":

A = 180° − (104.1° + 28.2°)
A = 180° − 132.3°
A = 47.7° to one decimal place

 

So A = 47.7°, B = 104.1°, and C = 28.2°

 

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