Isosceles Triangle Height Calculator
Calculate triangle heights to apex and from legs with step-by-step solutions
Calculate Isosceles Triangle Heights
Length of the two equal sides
Length of the unequal side
Height Calculation Results
Enter the required values to calculate triangle heights.
Example Calculation
Example: Roof Truss Height
Problem: A triangular roof truss has two equal sides of 15 cm each and a base of 10 cm. What are the heights?
Given: Leg length (a) = 15 cm, Base length (b) = 10 cm
Method: Using legs and base (Pythagorean theorem)
Solution
Check validity: 2a > b → 2(15) = 30 > 10 ✓
Height to apex: hb = √(a² - (b/2)²) = √(15² - 5²) = √(225 - 25) = √200 = 14.14 cm
Area: = ½ × 10 × 14.14 = 70.71 cm²
Height from leg: ha = (2 × 70.71) / 15 = 9.43 cm
Result: Height to apex = 14.14 cm, Height from leg = 9.43 cm
Height Formulas
Height to Apex (hb)
hb = √(a² - (b²/4))
From Pythagorean theorem
From Area
h = (2 × Area) / base
Direct from area formula
Height from Leg (ha)
ha = (2 × Area) / leg
Altitude to opposite vertex
From Angles
h = a × sin(angle)
Using trigonometry
Types of Heights
Height to Apex (hb)
Perpendicular from base to vertex between equal sides
Height from Leg (ha)
Perpendicular from leg endpoint to opposite vertex
Altitude Properties
Both heights intersect at the orthocenter
Triangle Validity
Sum of two legs > base (2a > b)
Each leg > half base (a > b/2)
All sides > 0
Vertex angle < 180°
Base angles < 90°
Understanding Isosceles Triangle Heights
What are Triangle Heights?
In an isosceles triangle, there are multiple heights (altitudes) that can be calculated. The most important are the height to the apex (from base to vertex) and the heights from the legs to their opposite vertices.
Key Mathematical Relationships
- •Pythagorean foundation: a² = hb² + (b/2)²
- •Area relationship: Area = ½ × base × height
- •Altitude formula: ha = (2 × Area) / leg
- •Trigonometric: h = a × sin(angle)
Calculation Methods
Pythagorean Method
Most reliable using the right triangle formed by height, leg, and half-base.
Area-Based Method
Direct calculation when area is known, using the area formula.
Trigonometric Method
Uses angles and trigonometric functions for height calculation.
Real-World Applications
Architecture & Construction
Calculate roof heights, triangular supports, and structural clearances in building design.
Engineering
Determine heights in bridge trusses, antenna towers, and mechanical component design.
Mathematics Education
Understand geometric relationships, verify triangle properties, and solve geometry problems.