Solving SSA Triangles

"SSA" means "Side, Side, Angle"

SSA Triangle

This means we are given two sides and an angle that is not the angle between the sides.

 

To solve an SSA triangle

 

Example 1

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:

sinC/c = sinB/b

sinC/13 = sin(31°)/8

So sinC = (13×sin(31°))/8

= 0.8369...

So C = sin-1(0.8369...)

= 56.818...°
= 56.8° correct to one decimal place. (*See below)

Next, we can use 'angles of a triangle' to find angle A:

A = 180° - 31° - 56.818...°

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

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

a/sinA = b/sinB

So a/sin(92.181...°) = 8/sin(31°)

Did you 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 you should still have on your calculator screen from the last calculation.

So a = (sin(92.181...°) × 8)/sin(31°)

= 15.52 correct to 2 decimal places

So, we have completely solved the triangle ...

... or have we?

Back when we calculated:

So C = sin-1(0.8369...)

= 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...°

So let's go back and continue our example:

The other possible angle is:

C = 180° - 56.818...°

= 123.181... = 123.2° correct to one decimal place

With a new value for C we should also re-calculate angle A and side a

Use 'angles of a triangle add to 180°' to find angle A:

A = 180° - 31° - 123.181...°

= 25.818...° = 25.8° correct to one decimal place.

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

a/sinA = b/sinB

So a/sin25.818...° = 8/sin31°

So a = (sin25.818...°×8)/sin31°

= 6.76 correct to 2 decimal places

Here you can see why we have two possible answers:

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

 

Example 2

This is also an SSA triangle.

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

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

sinL/l = sinM/m

sinL/7.6 = sin125°/12.4

So sinL = (7.6×sin125°)/12.4

= 0.5020...

So L = 30.136...°

= 30.1° correct to one decimal place.

Next, we will use 'angles of a triangle' to find angle N:

N = 180° - 125° - 30.136...°

= 24.863...° = 24.9° correct to one decimal place.

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

n/sinN = m/sinM

So n/sin24.863...° = 12.4/sin125°

So n = (sin24.863...°×12.4)/sin125°

= 6.36 correct to 2 decimal places

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

The other "possible" answer for L would be 149.9° which is impossible. We already have M = 125° and we can't have two obtuse angles in a triangle (they would add to more than 180°)

Conclusion:

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