Isosceles Triangle Calculator
Calculate area, perimeter, angles, and heights of isosceles triangles with two equal sides
Calculate Isosceles Triangle Properties
Length of the two equal sides
Length of the third side (different from legs)
Example Calculation
Example: Golden Triangle
Given: Leg length = 5 units, Base length = 3.09 units
This is approximately a golden triangle where a/b ≈ φ (golden ratio)
Step-by-Step Solution
1. Height from base: h = √(5² - 3.09²/4) = √(25 - 2.38) ≈ 4.76 units
2. Area: A = ½ × 3.09 × 4.76 ≈ 7.35 square units
3. Perimeter: P = 2 × 5 + 3.09 = 13.09 units
4. Vertex angle: β = arccos((2×25 - 3.09²)/(2×25)) ≈ 36°
5. Base angles: α = (180° - 36°)/2 = 72° each
Isosceles Triangle Properties
Two Equal Sides
Has two sides of equal length (legs)
Equal Base Angles
Angles opposite the equal sides are equal
Axis of Symmetry
Has a line of symmetry through the vertex
Quick Tips
Triangle inequality: Sum of any two sides > third side
Vertex angle + 2 × base angle = 180°
Equilateral triangle is a special isosceles triangle
Golden triangle has legs/base ratio = φ ≈ 1.618
Understanding Isosceles Triangles
What is an Isosceles Triangle?
An isosceles triangle is a triangle with two sides of equal length, called legs. The third side is called the base. The vertex angle is the angle between the legs, and the base angles are the angles adjacent to the base.
Key Properties
- •Has an axis of symmetry along the vertex height
- •The two base angles are equal
- •Can be acute, right, or obtuse (depends on vertex angle)
- •Equilateral triangle is a special case
Area Formulas
Given legs (a) and base (b):
Area = ¼ × b × √(4a² - b²)
Given height (h) and base (b):
Area = ½ × b × h
Given legs (a) and vertex angle (β):
Area = ½ × a² × sin(β)
Special Cases
Golden Triangle: Legs to base ratio equals golden ratio (φ ≈ 1.618)
Right Isosceles: Vertex angle = 90°, base angles = 45° each