SSA (Side, Side, Angle) is when we know 2 sides and 1 angle that is not the angle between the sides.
Table of Contents
How to solve an SSA triangle ?
First, use Law of Sines to calculate one of the other two angles.
Then, use the three angles add to 180° to find the other angle.
Finally use The Law of Sines again to find the unknown side.
Example
Find the unknown side and angles
Answer
From this triangle we get
angle A = 31°
a = 8 and
c = 13
In this case, we can use The Law of Sines first to find angle C:
sin(C)/c = sin(A)/a
sin(C)/13 = sin(31°)/8
sin(C) = (13×sin(31°))/8
sin(C) = 0.8369…
C = sin−1(0.8369…)
C = 56.818…°
C = 56.8° to one decimal place
Then, use the 3 angles add to 180° to find angle B:
B = 180° − 31° − 56.818…°
B = 92.181…° = 92.2° to one decimal place
Next, use The Law of Sines again to find b:
b/sin(B) = a/sin(A)
b/sin(92.181…°) = 8/sin(31°)
We didn’t use B = 92.2°, (the angle is rounded to 1 decimal place).
It’s better to use the unrounded number 92.181…° which still be on our calculator from the last calculation.
b = (sin(92.181…°) × 8)/sin(31°)
b = 15.52 to 2 decimal places
Other possible answer?
Back to:
C = sin−1(0.8369…)
C = 56.818…°
sin−1(0.8369…) have two answers
The other answer for C is 180° − 56.818…°
The other possible angle is:
C = 180° − 56.818…°
C = 123.2° to one decimal place
With a different value for C we will have different values for angle B and side b
Use the 3 angles add to 180° to find angle B:
B = 180° − 31° − 123.181…°
B = 25.818…°
B = 25.8° to one decimal place
Now we can use The Law of Sines again to find b:
b/sin(B) = a/sin(A)
b/sin(25.818…°) = 8/sin(31°)
b = (sin(25.818…°)×8)/sin(31°)
b = 6.76 to 2 decimal places
So the 2 sets of answers are:
C = 56.8°, B = 92.2°, b = 15.52
C = 123.2°, B = 25.8°, b = 6.76