Solving SSS Triangles - Math Is Fun

Solving SSS Triangles

"SSS" means "Side, Side, Side"

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

To solve an SSS triangle:

• use The Law of Cosines first to calculate one of the angles
• then use The Law of Cosines again to find another angle
• and finally use angles of a triangle add to 180° to find the last angle

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

cos(C) = a2 + b2 − c22ab

cos(A) = b2 + c2 − a22bc

cos(B) = c2 + a2 − b22ca

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

First check the three sides actually make a triangle! The two shorter sides together must be longer than the longest side.

Example 1

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 angle. 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 ... we have found all its angles.

The triangle can have letters other than ABC:

Example 2

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'll use The Law of Cosines again to find angle Y:

cos(Y) = (z2 + x2 − y2)/2zx cos(Y) = (3.52 + 5.12 − 7.92) / (2 × 3.5 × 5.1) 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's another (slightly faster) way to solve an SSS triangle:

• use the Law of Cosines first to calculate the largest angle
• then use the Law of Sines to find another angle
• and finally use angles of a triangle add to 180° to find the last angle

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

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°

All done! And just a little easier.

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