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Equilateral Triangle

An equilateral triangle has all three sides equal and all three angles 60°. One measurement (side, height, area, perimeter, inradius, or circumradius) determines all others. Perfect symmetry.

Concept Fundamentals
A = (√3/4)a²
Area
h = (√3/2)a
Height
60° each
All angles
Circum = 2×inradius
R = 2r

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Six equilateral triangles around a point form a regular hexagon — honeycomb geometry. All four notable centers (centroid, orthocenter, incenter, circumcenter) coincide. Equilateral triangles tessellate the plane with no gaps.

Key quantities
A = (√3/4)a²
Area
Key relation
h = (√3/2)a
Height
Key relation
60° each
All angles
Key relation
Circum = 2×inradius
R = 2r
Key relation

Ready to run the numbers?

Why: Equilateral triangles tessellate the plane, appear in honeycombs and crystals, and have perfect symmetry—centroid, orthocenter, incenter, and circumcenter all coincide.

How: From side: h = (√3/2)a, A = (√3/4)a², P = 3a, r = a/(2√3), R = a/√3. From any other: work backwards to find side, then compute the rest.

Six equilateral triangles around a point form a regular hexagon — honeycomb geometry.All four notable centers (centroid, orthocenter, incenter, circumcenter) coincide.
Sources:Wolfram MathWorld - EquilateralKhan Academy - Geometry

Run the calculator when you are ready.

Equilateral Triangle CalculatorEnter side, height, area, perimeter, inradius, or circumradius to find all properties
🔺
PERFECT SYMMETRY

Equilateral Triangle — All Sides Equal, All Angles 60°

One measurement unlocks everything. Area, perimeter, height, inradius, circumradius — all derived from the √3 constant.

Triangle Area →Right Triangle →Perimeter →

🔺 Sample Examples — Click to Load

Input Configuration

Equilateral Triangle Visualization

equilateral_triangle.sh
CALCULATED
$ calc_equilateral --input=side --value=10.0000
Side
10.0000
units
Area
43.3013
sq units
Perimeter
30.0000
units
Height
8.6603
units
Inradius
2.8868
units
Circumradius
5.7735
units
Angles
60° / 60° / 60°
R/r ratio
2.0000
Share:
Equilateral Triangle Calculation
Side 10.0000 → Area 43.3013 sq units
43.3013 sq units
P = 30.0000h = 8.6603r = 2.8868R = 5.7735
numbervibe.com/calculators/mathematics/triangle/equilateral

Equilateral Triangle Properties Radar

Property Comparison (Bar)

Property Proportions (Doughnut)

Step-by-Step Breakdown

DEFINITION
Equilateral triangle property
All sides = a, all angles = 60°
ext{Perfect} ext{symmetry}
INPUT
Given side length
a = 10.0000
ext{Direct} ext{input}
DERIVED
Area formula
A = (√3/4)·a²
ext{Standard} ext{equilateral} ext{area}
RESULT
AREA
43.3013 sq units
Perimeter
P = 3a = 30.0000 units
ext{Sum} ext{of} ext{three} ext{equal} ext{sides}
Height
h = (√3/2)·a = 8.6603 units
ext{Altitude} ext{from} ext{any} ext{vertex}
Inradius
r = a/(2√3) = 2.8868 units
ext{Inscribed} circle ext{radius}
Circumradius
R = a/√3 = 5.7735 units
ext{Circumscribed} circle ext{radius}
Angles
60° / 60° / 60°
ext{All} ext{angles} ext{equal} ext{in} ext{equilateral}

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🔺

Equilateral triangles are the only triangles where the four notable centers coincide.

— Wolfram MathWorld

📐

Six equilateral triangles around a point form a regular hexagon — honeycomb symmetry.

— Khan Academy

Key Takeaways

  • An equilateral triangle has all three sides equal and all three angles equal to 60°
  • Knowing just one measurement (side, height, area, perimeter, inradius, or circumradius) determines all others
  • The height is always h = (√3/2) × side — about 0.866 times the side length
  • Area = (√3/4) × side² — the circumradius is exactly twice the inradius (R = 2r)
  • Equilateral triangles tessellate the plane perfectly and appear in honeycombs, crystals, and trusses

Did You Know?

🔺Equilateral triangles are the only triangles where the centroid, orthocenter, incenter, and circumcenter all coincide at the same point — perfect symmetry in every waySource: Wolfram MathWorld
📐Six equilateral triangles arranged around a point form a regular hexagon — that's why honeycombs and snowflakes have hexagonal symmetrySource: Khan Academy
🏗️Triangular trusses in bridges and roofs often use equilateral triangles because they distribute stress evenly and cannot deform without changing side lengthsSource: Engineering Toolbox
🧪The BF₃ molecule has a trigonal planar structure with 120° angles — three equilateral triangles sharing a centerSource: Chemistry LibreTexts
🎨M.C. Escher and Islamic tile artists used equilateral triangles in tessellations — shapes that tile the plane with no gapsSource: Math Art
📡GPS triangulation uses triangles to pinpoint location — equilateral arrangements minimize geometric dilution of precisionSource: NASA

How Equilateral Triangle Formulas Work

Because all sides and angles are equal, one known value unlocks every other property. Each formula derives from the fundamental relationship between side length and the √3 constant.

From Side to Everything

Given side length aa: Area = (√3/4)a², Height = (√3/2)a, Perimeter = 3a, Inradius r = a/(2√3), Circumradius R = a/√3. The ratio R/r = 2 for all equilateral triangles. Use our Triangle Area Calculator for general triangles.

Reverse Engineering from Area or Height

If you know area A: side = √(4A/√3). If you know height h: side = (2/√3)h. These inversions let you work backward from any measurement. See the Triangle Height Calculator for other triangle types.

Circle Radii Relationships

The inradius (inscribed circle) touches all three sides; the circumradius (circumscribed circle) passes through all three vertices. In an equilateral triangle, R = 2r always. This is unique among triangles. Check Right Triangle Calculator for comparison.

Expert Tips for Equilateral Triangles

Height ≠ Side Length

A common mistake: the height is (√3/2) × side ≈ 0.866 × side, not equal to the side. The height is always shorter than the side.

Don't Confuse Inradius and Circumradius

Inradius r is the circle inside; circumradius R is the circle through the vertices. R = 2r for equilateral triangles — memorize this ratio.

Use the Right Input Type

If you measured the altitude, select "Height." If you have the area from a planimeter, select "Area." Choosing correctly avoids extra conversion steps.

Tessellation Applications

Equilateral triangles tile the plane with no gaps — useful for floor patterns, quilting, and structural grids. Six form a hexagon.

Why Use This Calculator vs. Other Tools?

FeatureThis CalculatorWolfram AlphaManual Calculation
6 input types (side, height, area, P, r, R)✅✅⚠️ Tedious
Step-by-step solutions✅⚠️ Paid❌
Interactive visualization✅✅❌
Charts (Radar, Bar, Doughnut)✅❌❌
Copy & share results✅❌❌
AI-powered explanation✅❌❌
500ms debounce auto-calc✅❌❌
7 sample examples✅❌❌
Free (no signup)✅⚠️ Limited✅

Frequently Asked Questions

Why are all angles 60° in an equilateral triangle?

The sum of angles in any triangle is 180°. With three equal angles, each must be 180° ÷ 3 = 60°. This is a direct consequence of the triangle angle sum theorem.

Can an equilateral triangle be a right triangle?

No. A right triangle has one 90° angle. An equilateral triangle has three 60° angles. The two cannot be the same shape.

What is the relationship between side and height?

Height h = (√3/2) × side ≈ 0.866 × side. The height is always about 86.6% of the side length. Derive it using the Pythagorean theorem on half the triangle.

Why is the circumradius twice the inradius?

For equilateral triangles, R = a/√3 and r = a/(2√3), so R/r = 2. The circumscribed circle has exactly twice the radius of the inscribed circle — a unique property.

How do I find the side if I only know the area?

Use a = √(4A/√3). For example, if A = 100, then a = √(400/√3) ≈ 15.2. Or use this calculator with input type "Area."

What is tessellation and why do equilateral triangles tessellate?

Tessellation means tiling a plane with no gaps. Equilateral triangles fit together perfectly because six of them form a regular hexagon, and hexagons tile the plane. Bees use this in honeycombs.

Where are equilateral triangles used in real life?

Trusses in bridges and roofs, geodesic domes, traffic signs, corporate logos (e.g., Delta, Mitsubishi), quilting patterns, molecular structures (e.g., BF₃), and crystallography.

How accurate is this calculator?

It uses IEEE 754 double-precision arithmetic (~15 significant digits). Results are rounded to 4 decimal places. For most engineering and educational uses, this exceeds required precision.

Equilateral Triangle by the Numbers

60°
Each Angle
√3/4
Area Factor
R = 2r
Circum/Inradius
6
Form a Hexagon

Disclaimer: This calculator provides mathematically precise results based on standard equilateral triangle formulas. Results are limited by floating-point precision (~15 significant digits). For critical engineering or scientific applications, always verify with domain-specific tools. Not a substitute for professional surveying or structural analysis.

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