Solving SSA Triangles

SSA (Side, Side, Angle) is when we know 2 sides and 1 angle that is not the angle between the sides.

SSA triangle

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

SSA triangle 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°)

Note:

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

Solution

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

Learn More

Solving SAS Triangles

Solving SSS Triangles

Finding an Angle in a Right Angled Triangle

Vertical Angles

Algebra Index

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