Isosceles Triangle Angles Calculator
Calculate vertex angle and base angles of isosceles triangles with step-by-step solutions
Calculate Isosceles Triangle Angles
Length of the two equal sides
Length of the third side
Angle Calculation Results
Enter the required values to calculate the triangle angles.
Example Calculation
Example: Roof Truss Design
Problem: A roof truss forms an isosceles triangle with rafters (legs) of 5m each and a base span of 6m.
Given: Leg length (a) = 5m, Base length (b) = 6m
Find: All angles of the triangle
Solution
Step 1: Calculate vertex angle using Law of Cosines
Formula: cos(β) = (2a² - b²) / (2a²)
Substitute: cos(β) = (2×5² - 6²) / (2×5²) = (50 - 36) / 50 = 0.28
Calculate: β = arccos(0.28) = 73.7°
Step 2: Calculate base angles
Formula: α = (180° - β) / 2 = (180° - 73.7°) / 2 = 53.2°
Result: Vertex angle = 73.7°, Base angles = 53.2° each
Angle Formulas
Vertex Angle from Sides
cos(β) = (2a² - b²) / (2a²)
Base Angles
α = (180° - β) / 2
Angle Sum
α + α + β = 180°
Base from Vertex Angle
b = 2a × sin(β/2)
Triangle Types
Acute Isosceles
Vertex angle < 90°
Right Isosceles
Vertex angle = 90°, Base angles = 45°
Obtuse Isosceles
Vertex angle > 90°
Equilateral
All angles = 60°
Isosceles Triangle Properties
Two sides are equal (legs)
Two base angles are equal
Has one line of symmetry
Height bisects vertex angle and base
Sum of all angles = 180°
Understanding Isosceles Triangle Angles
What Makes This Calculator Special?
This calculator is specifically designed for isosceles triangles and offers three different calculation methods. It takes advantage of the special properties of isosceles triangles, where two sides are equal and two angles are equal.
Key Angle Relationships
- •Base angles are equal: The two angles adjacent to the base are always identical
- •Angle sum is 180°: α + α + β = 180°, so 2α + β = 180°
- •Vertex angle uniqueness: The vertex angle determines the entire triangle shape
- •Special cases: When vertex = 90°, base angles = 45° each
Mathematical Methods
From Side Lengths
Uses the Law of Cosines to find the vertex angle, then calculates base angles using the angle sum property: cos(β) = (2a² - b²) / (2a²)
From Angles
Uses the relationship that base angles are equal and sum to 180° with the vertex angle: α = (180° - β) / 2
Trigonometric Relations
Connects angles and sides using sine and cosine functions for complete triangle solutions.
Real-World Applications
Architecture & Construction
Calculate roof angles, truss designs, and architectural elements with symmetric triangular shapes.
Engineering
Design bridges, support structures, and mechanical components with triangular reinforcements.
Education
Learn trigonometry, geometry concepts, and solve homework problems with step-by-step solutions.