Solving SSA Triangles

"SSA" means "Side, Side, Angle"

SSA Triangle

"SSA" is when we know two sides and an angle that is not the angle between the sides.

Example 1

trig SSA example

In this triangle we know

  • angle B = 31°
  • b = 8
  • and c = 13

In this case, we can use The Law of Sines first to find angle C:

sin(C)/c = sin(B)/b

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 (*see below)

Next, we can use the three angles add to 180° to find angle A:

A = 180° − 31° − 56.818...°

A = 92.181...° = 92.2° to one decimal place

Now we can use The Law of Sines again to find a:

a/sin(A) = b/sin(B)

a/sin(92.181...°) = 8/sin(31°)

Notice that we didn't use A = 92.2°, that angle is rounded to 1 decimal place. It's much better to use the unrounded number 92.181...° which should still be on our calculator from the last calculation.

a = (sin(92.181...°) × 8)/sin(31°)

a = 15.52 to 2 decimal places

So, we have completely solved the triangle ...

... or have we?

* Back when we calculated:

C = sin−1(0.8369...)
C = 56.818...°

We didn't think that sin−1(0.8369...) might have two answers (see Law of Sines)

The other answer for C is 180° − 56.818...°

Here you can see why we have two possible answers:

trig SSA example

By swinging side "8" left and right we can
join up with side "a" in two possible locations.

So let's go back and continue our example:

The other possible angle is:

C = 180° − 56.818...°

C = 123.2° to one decimal place

With a new value for C we will have new values for angle A and side a

Use "the three angles add to 180°" to find angle A:

A = 180° − 31° − 123.181...°

A = 25.818...°

A = 25.8° to one decimal place

Now we can use The Law of Sines again to find a:

a/sin(A) = b/sin(B)

a/sin(25.818...°) = 8/sin(31°)

a = (sin(25.818...°)×8)/sin(31°)

a = 6.76 to 2 decimal places

So the two sets of answers are:

C =  56.8°, A = 92.2°, a = 15.52

C = 123.2°, A = 25.8°, a = 6.76

Example 2

trig SSA example

This is also an SSA triangle.

In this triangle we know angle M = 125°, m = 12.4 and l = 7.6

We will use The Law of Sines to find angle L first:

sin(L)/l = sin(M)/m

sin(L)/7.6 = sin(125°)/12.4

sin(L) = (7.6×sin(125°))/12.4

sin(L) = 0.5020...

L = 30.136...°

L = 30.1° to one decimal place

Next, we will use "the three angles add to 180°" to find angle N:

N = 180° − 125° − 30.136...°

N = 24.863...°

N = 24.9° to one decimal place

Now we will use The Law of Sines again to find n:

n/sin(N) = m/sin(M)

n/sin(24.863...°) = 12.4/sin(125°)

n = (sin(24.863...°)×12.4)/sin(125°)

n = 6.36 to 2 decimal places

trig SSA example

Note there is only one answer in this case. The "12.4" line only joins up one place.

The other possible answer for L is 149.9°. But that is impossible because we already have M = 125° and a triangle can't have two angles greater than 90°.

Conclusion:

When solving a "Side, Side, Angle" triangle we need to
check if there could be another possible answer!