Triangle Sum Theorem Calculator
Find missing angles using the triangle angle sum theorem (180°)
Triangle Angle Calculation
Missing Angle Results
Triangle Sum Theorem Examples
Example 1: Find Missing Angle
Given: Angle A = 40°, Angle B = 75°
Find: Angle C
Solution: C = 180° - 40° - 75° = 65°
Verification: 40° + 75° + 65° = 180° ✓
Example 2: Right Triangle
Given: Angle A = 30°, Angle B = 60°
Find: Angle C
Solution: C = 180° - 30° - 60° = 90°
Type: Right Triangle (has 90° angle)
Example 3: Using Radians
Given: Angle A = π/4 rad, Angle B = π/3 rad
Find: Angle C
Solution: C = π - π/4 - π/3 = 5π/12 rad
In degrees: C ≈ 75°
Triangle Sum Theorem
The Fundamental Rule
α + β + γ = 180°
α + β + γ = π rad
Finding Missing Angle
α = 180° - β - γ
β = 180° - α - γ
γ = 180° - α - β
Triangle Classification
Acute Triangle
All angles < 90°
Right Triangle
One angle = 90°
Obtuse Triangle
One angle > 90°
Quick Tips
The sum is always exactly 180° (or π radians)
Each angle must be greater than 0°
Each angle must be less than 180°
Use this theorem to check triangle validity
Understanding the Triangle Sum Theorem
What is the Triangle Sum Theorem?
The triangle sum theorem is one of the most fundamental principles in geometry. It states that the sum of all interior angles in any triangle always equals 180 degrees (or π radians).
Mathematical Proof
This theorem can be proven by drawing a line parallel to one side of the triangle through the opposite vertex, creating alternate interior angles that demonstrate the 180° sum.
Applications
- •Finding missing angles in triangles
- •Validating triangle measurements
- •Solving geometry problems
- •Architecture and engineering
Step-by-Step Process
Step 1: Identify Known Angles
Determine which angles you know and which one you need to find.
Step 2: Apply the Theorem
Missing Angle = 180° - Known Angle 1 - Known Angle 2
Step 3: Verify the Result
Check that all three angles sum to exactly 180°.
Unit Conversions
Degrees to Radians: rad = deg × π/180
Radians to Degrees: deg = rad × 180/π