Equilateral Triangle Calculator

What is an Equilateral Triangle?

An equilateral triangle is a special type of triangle where all three sides have equal length and all three angles are equal, each measuring 60°. The word "equilateral" comes from the Latin words "aequus" (equal) and "latus" (side). This unique geometry gives the equilateral triangle perfect symmetry and many interesting properties.

Key Properties of Equilateral Triangles

• All three sides have equal length
• All three angles are equal, each measuring 60°
• Has three axes of symmetry
• The centroid, orthocenter, and circumcenter coincide at the same point

Formulas for Equilateral Triangles

Let \(a\) be the length of each side. Then:

1. Area: \(A = \frac{\sqrt{3}}{4}a^2\)
2. Perimeter: \(P = 3a\)
3. Height (altitude): \(h = \frac{\sqrt{3}}{2}a\)
4. Inradius (radius of inscribed circle): \(r = \frac{a}{2\sqrt{3}}\)
5. Circumradius (radius of circumscribed circle): \(R = \frac{a}{\sqrt{3}}\)

Step-by-Step Calculations

Let's calculate these properties for an equilateral triangle with side length \(a = 6\) units:

1. Area: \[A = \frac{\sqrt{3}}{4}a^2 = \frac{\sqrt{3}}{4}(6^2) = 9\sqrt{3} \approx 15.59 \text{ square units}\]
2. Perimeter: \[P = 3a = 3(6) = 18 \text{ units}\]
3. Height: \[h = \frac{\sqrt{3}}{2}a = \frac{\sqrt{3}}{2}(6) = 3\sqrt{3} \approx 5.20 \text{ units}\]
4. Inradius: \[r = \frac{a}{2\sqrt{3}} = \frac{6}{2\sqrt{3}} = \frac{\sqrt{3}}{2} \approx 1.73 \text{ units}\]
5. Circumradius: \[R = \frac{a}{\sqrt{3}} = \frac{6}{\sqrt{3}} = 2\sqrt{3} \approx 3.46 \text{ units}\]

Visual Representation

This diagram illustrates the equilateral triangle with the calculated dimensions, including the inscribed and circumscribed circles.