Isosceles Triangle Side Calculator
Calculate missing sides and properties of isosceles triangles using various input combinations
Calculate Isosceles Triangle Sides
Length of the equal sides
Length of the unique side
Triangle Properties
Perimeter: 0.00 units
Triangle Type: Isosceles
Equal sides: 2 (legs)
Equal angles: 2 (base angles)
Example Calculation
Find Base Length
Given: Leg length a = 10 units, Base angle α = 30°
Formula: b = 2 × a × cos(α)
Calculation: b = 2 × 10 × cos(30°) = 2 × 10 × 0.866 = 17.32 units
Find Leg Length
Given: Base length b = 12 units, Height h = 8 units
Formula: a = √(h² + (b/2)²)
Calculation: a = √(8² + (12/2)²) = √(64 + 36) = √100 = 10 units
Isosceles Triangle Properties
Two Equal Sides
The legs (a) are equal in length
Two Equal Angles
Base angles (α) are equal
Symmetrical
Height bisects the base and vertex angle
Key Formulas
b = 2a × cos(α)
Base from leg and angle
a = √(h² + (b/2)²)
Leg from base and height
h = a × sin(α)
Height from leg and angle
Area = ½ × b × h
Triangle area
Understanding Isosceles Triangle Calculations
What is an Isosceles Triangle?
An isosceles triangle is a triangle with two sides of equal length, called legs (a). The third side is called the base (b). The angles opposite to the equal sides (base angles α) are also equal, while the vertex angle β is unique.
Key Properties
- •Two equal sides (legs): a = a
- •Two equal base angles: α = α
- •Sum of all angles: 2α + β = 180°
- •Height creates two congruent right triangles
Calculation Methods
From Two Sides
When you know both legs and base, use the Pythagorean theorem to find height: h = √(a² - (b/2)²)
From Side and Angle
Use trigonometry: if you know a leg and base angle, the base equals 2a × cos(α)
From Side and Height
The height splits the triangle into two right triangles, allowing use of the Pythagorean theorem
Triangle Inequality Theorem
For a valid isosceles triangle, the sum of any two sides must be greater than the third side:
- • a + a > b → 2a > b
- • a + b > a → b > 0 (always true for positive values)
- • a + b > a → b > 0 (always true for positive values)
This means the leg length must be greater than half the base length: a > b/2