# Equiangular Triangles: Definition, Properties, and Examples

TL;DR

An equiangular triangle is a triangle whose three interior angles are all equal, which forces each angle to be 60° and makes it identical to an equilateral triangle. This article covers equiangular triangles in full: the definition, properties, the area and perimeter formulas, why equal angles guarantee equal sides, and the mistakes students make most often.

## What is an Equiangular Triangle

An **equiangular triangle** is a triangle in which all three interior angles are equal. Because the three angles of any triangle add to 180°, three equal angles must each measure \(\frac{180°}{3} = 60°\). An equiangular triangle is also a 60°–60°–60° triangle.

The word breaks down cleanly: _equi-_ means "equal" and _-angular_ means "angles." A triangle that is equal in its angles is equiangular. The single most important fact follows directly — in a triangle, equal angles force equal sides, so an equiangular triangle is exactly the same shape as an [equilateral triangle](/content/math/geometry/equilateral-triangle/index.html).

## Why Does Equal Angles Mean Equal Sides?

If all three angles are equal, then no side can be opposite a "bigger" angle than another — so all three sides must match. That single rule is why we never need to measure the sides of an equiangular triangle separately; the angles already settle it.

## Properties of An Equiangular Triangle

An equiangular triangle carries every property of an equilateral triangle, plus the angle facts that name it. The core properties:

- **Every interior angle is 60°.** This is fixed; it can never be anything else.
- **All three sides are equal in length.** Equal angles force equal sides.
- **It is always an acute triangle.** Since the largest angle is only 60°, no angle reaches 90°, so the triangle can never be right-angled or obtuse.
- **Each exterior angle is 120°.** An interior angle of 60° leaves 180°−60°=120° on the outside.
- **The centre points coincide.** The centroid, orthocentre, circumcentre, and incentre all land on the same single point — a level of symmetry no other triangle has.
- **It has three lines of symmetry**, one through each vertex and the midpoint of the opposite side.

## The Formulas for An Equiangular Triangle

Because an equiangular triangle is fully determined by a single side length \(a\), every measurement follows from that one number. Here is what each formula means and where it comes from.

| Quantity                   | Formula                         | What it gives you                                    |
| -------------------------- | ------------------------------- | --------------------------------------------------- |
| Perimeter                  | \(P = 3a\)                    | Three equal sides, so just add one side three times  |
| Area                       | \(A = \frac{\sqrt{3}}{4} a^2\) | The space enclosed                                  |
| Height (altitude)         | \(h = \frac{\sqrt{3}}{2} a\) | The straight-line drop from a vertex to the opposite side |

## Examples of Equiangular Triangles

### Example 1
**An equiangular triangle has one side measuring 8 cm. Find its perimeter.**

\[P = 3a = 3 \times 8 = 24 \text{ cm}\]

**Final answer:** The perimeter is 24 cm.

### Example 2
**A triangle has angles 60°, 60°, and 60°. A student claims you cannot find the third side without measuring it. Is the student right?**

**Final answer:** No, the student is wrong. In an equiangular triangle, knowing one side determines all three.

### Example 3
**Find the area of an equiangular triangle with side length 6 cm.**

\[A = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \approx 15.59 \text{ cm}^2\]

**Final answer:** The area is approximately 15.59 cm².

### Example 4
**What is the measure of each exterior angle?**

Each exterior angle is 180°−60°=120°, and the three exterior angles add to 360°.

**Final answer:** Each exterior angle is 120°, and the three together total 360°.

### Example 5
**Find the height of an equiangular triangle whose side is 10 cm.**

\[h = \frac{\sqrt{3}}{2} \times 10 \approx 8.66 \text{ cm}\]

**Final answer:** The height is approximately 8.66 cm.

### Example 6
**The area of an equiangular triangle is 16√3 cm². Find the length of one side.**

**Final answer:** Each side is 8 cm.

## Conclusion

- An **equiangular triangle** has three equal interior angles, each measuring exactly 60°.
- Equal angles force equal sides, so an equiangular triangle is the same figure as an equilateral triangle.
- It is always an acute triangle, with each exterior angle equal to 120°.
- Its area is \(A = \frac{\sqrt{3}}{4} a^2\) and its perimeter is \(P = 3a\) — both set by a single side length \(a\).

## The Mistakes Students Make Most Often

1. **Treating equiangular and equilateral as different shapes.** 
2. **Assuming "equiangular" extends to all polygons the same way.** 
3. **Misusing the area formula by squaring the wrong quantity.**