SSS Triangle Calculator
Solve Side-Side-Side triangles using the Law of Cosines and Heron's formula
Calculate SSS Triangle
First side length
Second side length
Third side length
Triangle Inequality Check
Example Calculation
Given: a = 3 cm, b = 4 cm, c = 5 cm
Step 1: Check triangle inequality
3 + 4 = 7 > 5 ✓, 3 + 5 = 8 > 4 ✓, 4 + 5 = 9 > 3 ✓
Step 2: Calculate angles using Law of Cosines
cos(A) = (4² + 5² - 3²) / (2×4×5) = (16 + 25 - 9) / 40 = 0.8
A = arccos(0.8) = 36.87°
Step 3: Calculate area using Heron's formula
s = (3 + 4 + 5) / 2 = 6
Area = √(6×(6-3)×(6-4)×(6-5)) = √(6×3×2×1) = 6 cm²
Triangle Classification
Acute Triangle
All angles < 90°
Right Triangle
One angle = 90°
Obtuse Triangle
One angle > 90°
Key Formulas
Law of Cosines
cos(A) = (b² + c² - a²) / (2bc)
Heron's Formula
Area = √(s(s-a)(s-b)(s-c))
where s = (a+b+c)/2
Triangle Inequality
a + b > c
Quick Tips
SSS means all three sides are known
Check triangle inequality before calculating
Use Law of Cosines to find angles
SSS triangles are always congruent
Understanding SSS Triangles
What is an SSS Triangle?
An SSS triangle is one where all three sides are known. This is the most straightforward triangle configuration to solve, as there is exactly one unique triangle that can be formed with three given side lengths (if they satisfy the triangle inequality).
Triangle Inequality Theorem
Before solving an SSS triangle, we must verify that the three sides can actually form a triangle:
- •The sum of any two sides must be greater than the third side
- •This must be true for all three combinations: a+b>c, a+c>b, b+c>a
Solving Steps
Step 1: Verify Triangle Inequality
Check that a+b>c, a+c>b, and b+c>a
Step 2: Apply Law of Cosines
Find each angle using cos(A) = (b²+c²-a²)/(2bc)
Step 3: Calculate Area
Use Heron's formula or Area = ½ab sin(C)
Step 4: Find Additional Properties
Calculate heights, radii, and classify triangle type
Why SSS Triangles Are Special
Congruence:
- • SSS triangles are always congruent
- • If three sides match, the triangles are identical
- • This is the SSS congruence criterion
Uniqueness:
- • Only one triangle can be formed with three given sides
- • No ambiguous cases (unlike SSA)
- • Always produces a unique solution