SSS Triangle – Formula, Theorem, Solved Examples - Math Monks

What are SSS Triangles

SSS meaning ‘Side-Side-Side’ are triangles where all three sides are known. They are used when we need to find the missing angles. Shown below is an SSS triangle, △ABC,with side lengths a, b, and c respectively.

Solving SSS Triangle with Formulas

Using Law of Cosines

It involves three steps:

Step 1: Use the Law of Cosines first to calculate one of the angles

Step 2: Use the Law of Cosines again to find the second angle

Step 3: Use the angle sum rule of a triangle to find the third angle

The Law of cosines formula (in angle version) for the above SSS triangle having sides a, b, and c and their corresponding angles, ∠A, ∠B, and ∠C is given below:

cos (C) = a2 + b2-c2/2ab

Similarly, the values for the other two angles can be determined using the formulas:

cos (A) = b2 + c2-a2/2bc

cos (B) = c2 + a2-b2/2ca

Let us take an example to understand the concept better.

Solved Example

Find the missing angles in the given SSS triangle.

Solution:

In the given triangle, the three sides are:a = 3 cm,b = 4 cm,c = 5 cmStep 1:Using the law of cosines, we will find any one angle of the given triangle.Let us find ∠A first,cos (A) = b2 + c2 – a2/2bc, here a = 3 cm, b = 4 cm, c = 5 cmcos (A) = (42 + 52 – 32)/(2 x 4 x 5)cos (A) = 32/40A = cos– 1 (0.8) = 36.86°Step 2:Next, we will find the second anglecos (B) = c2 + a2 – b2/2ca, here a = 3 cm, b = 4 cm, c = 5 cmcos (B) = (52 + 32 – 42)/(2 x 5 x 3)cos (B) = 18/30B = cos– 1 (0.6) = 53.13°Step 3:Finally, we will find the third angle; angle C, by using the angle sum rule.∠A + ∠B + ∠C = 180°, here ∠A = 36.86°, ∠B = 53.13°36.86°+ 53.13°+ ∠C = 180°∠C = 180° – (36.86°+ 53.13°)∠C = 90.01 degree

Alternative Method – Using Law of Cosines and Law of Sines

The other way to solve an SSS triangle is done using three steps:

Step 1: Use the Law of Cosines to calculate the largest angle

Step 2: Use the Law of Sines to find the second angle

Step 3: Use the angle sum rule of a triangle to find the third angle

The largest angle is calculated first as the other two angles are acute. This helps the Law of Sines to give accurate result.

Apart from the Law of Cosines discussed above, here we will use the Law of Sines.

If a, b, c are the three sides and A, B, and C are the three angles of a triangle, the Law of Sines is given by:

a/sin A = b/sin B = c/sin C

Solved Example

Find the missing angles in the given triangle.

Solution:

The given triangle is an SSS triangle, where the three sides are:a = 11.6 cm,b = 15.2 cm,c = 7.4 cmStep 1:Using the law of cosines, we will find the largest angle of the given triangle.cos (B) = c2 + a2 – b2/2ca, here a = 11.6  cm, b = 15.2  cm, c = 7.4  cmcos (B) = (7.42 + 11.62 – 15.22)/(2 x 11.6 x 7.4)cos (B) = (134.56 + 54.76 − 231.04)/171.68B = cos– 1 (- 0.24) = 103.88°Step 2:Next, we will find the second angle using the Law of Sines,b/sin B = c/sin C, here b = 15.2  cm, ∠B = 103.88°, c = 7.4 sin C = c x sin B/ bsin C =7.4 x sin 103.88°/15.2sin C = 0.47C = sin– 1 (0.24) = 28.2°Step 3:Finally, we will find the third angle; angle A, by using the angle sum rule.∠A + ∠B + ∠C = 180°, here ∠B = 103.88°, ∠C = 28.2°∠A + 103.88° + 28.2° = 180°∠A = 180° – (103.88° + 28.2°)∠A = 47.92°

SSS Triangle Congruence Theorem

Let us prove the above theorem in two different instances.

Prove SSS Triangle Congruence Theorem

Proof: 1

To prove: △ABC ≅△DBC

Proof:

Steps	Statements	Reasons
1.	AB ≅ DB AC ≅ DC	Given
2.	BC ≅ BC	Reflexive property of congruence
3.	△ABC ≅△DBC	SSS postulate (Hence proved)

Proof: 2

To prove: △ACB ≅△ACD

Proof:

Steps	Statements	Reasons
1.	AB ≅ AD C is the midpoint of BD	Given
2.	BC ≅ DC	Definition of midpoint
AC ≅ AC	Reflexive property of congruence
3.	△ACB ≅△ACD	SSS postulate (Hence proved)

Identify which pair of triangles below illustrates an SSS relationship.

Solution:

a) congruent by SSS, b) congruent by SSS, c) congruent by SSS, d) congruent by SSS.