Triangle Length Calculator
Calculate missing sides and angles of triangles using the Law of Cosines and Law of Sines
Triangle Calculator
Example Calculation
Find Missing Side (SAS)
Given: Side a = 3, Side b = 4, Angle γ = 90°
Find: Side c
Solution:
Using Law of Cosines: c² = a² + b² - 2ab×cos(γ)
c² = 3² + 4² - 2×3×4×cos(90°)
c² = 9 + 16 - 0 = 25
c = 5
Find Missing Angles (SSS)
Given: a = 3, b = 4, c = 5
Find: All angles
Using Law of Cosines: cos(α) = (b² + c² - a²) / (2bc)
cos(α) = (16 + 25 - 9) / (2×4×5) = 32/40 = 0.8
α = 36.87°, β = 53.13°, γ = 90°
Triangle Laws
Law of Cosines
c² = a² + b² - 2ab×cos(C)
For finding sides when you know two sides and included angle
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)
For finding angles and sides in various configurations
Angle Sum
A + B + C = 180°
The sum of all angles in a triangle
Triangle Types
Right Triangle
One angle = 90°
Obtuse Triangle
One angle > 90°
Acute Triangle
All angles < 90°
Equilateral
All sides equal
Isosceles
Two sides equal
Understanding Triangle Length Calculations
What is the Law of Cosines?
The Law of Cosines is a generalization of the Pythagorean theorem that applies to all triangles, not just right triangles. It relates the sides of a triangle to the cosine of one of its angles.
When to Use It?
- •When you know two sides and the included angle (SAS)
- •When you know all three sides and need to find angles (SSS)
- •For non-right triangles where Pythagorean theorem doesn't apply
Mathematical Formulas
Law of Cosines:
c² = a² + b² - 2ab×cos(C)
b² = a² + c² - 2ac×cos(B)
a² = b² + c² - 2bc×cos(A)
Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C)
Note: This calculator uses both laws to provide complete triangle solutions.