Equilateral Triangle Calculator
Calculate all properties of an equilateral triangle: area, height, perimeter, circumradius, and inradius
Calculate Equilateral Triangle Properties
Length in units
Example: Traffic Yield Sign
Problem
A standard traffic yield sign is an equilateral triangle with 36-inch sides. Calculate all its properties.
Solution
Given: Side length a = 36 inches
Height: h = (36 × √3) / 2 = 31.18 inches
Area: A = (36² × √3) / 4 = 561.18 in²
Perimeter: P = 3 × 36 = 108 inches
Circumradius: R = (36 × √3) / 3 = 20.79 inches
Inradius: r = (36 × √3) / 6 = 10.39 inches
Equilateral Triangle Properties
Sides & Angles
- • All three sides are equal
- • All three angles are 60°
- • Sum of angles = 180°
Special Lines
- • Altitude = Median = Angle bisector
- • All special lines coincide
- • Three lines of symmetry
Circles
- • Circumcenter = Incenter = Centroid
- • R = 2r (circumradius = 2 × inradius)
- • All centers coincide
Formula Reference
Area
A = (a² × √3) / 4
Height
h = (a × √3) / 2
Perimeter
P = 3a
Circumradius
R = (a × √3) / 3
Inradius
r = (a × √3) / 6
Understanding Equilateral Triangles
What is an Equilateral Triangle?
An equilateral triangle is a regular triangle where all three sides are equal in length. It's also called a regular triangle and has several unique properties that make it special among all triangles.
Key Properties
- •All three sides are equal (a = b = c)
- •All three angles are equal (60° each)
- •It has 3 lines of symmetry
- •Circumcenter, incenter, centroid, and orthocenter all coincide
Formula Derivations
Area Formula
Starting with: Area = ½ × base × height
Height from Pythagorean theorem:
h² + (a/2)² = a²
h = √(a² - a²/4) = a√3/2
Area = (a² × √3) / 4
Radius Formulas
For any triangle: R = abc/(4×Area)
Since a = b = c: R = a³/(4×Area)
R = a³/(4 × a²√3/4) = a/√3
R = (a × √3) / 3
r = Area/s = (a√3/4)/(3a/2) = (a × √3) / 6
Real-World Applications
Traffic Signs
Yield signs are equilateral triangles for high visibility and standardization.
Architecture
Triangular trusses and roof structures often use equilateral triangles for stability.
Engineering
Equilateral triangular patterns provide maximum strength with minimum material.