Equilateral & Equiangular Triangles
Equilateral and equiangular triangles are two descriptions of the same triangle: all sides equal ⟺ all angles equal (= 60°). This lesson proves this equivalence and explores its applications.
Introduction
The equilateral triangle is a “perfect” triangle — maximum symmetry, every side and every angle identical. It's a corollary of the Isosceles Triangle Theorem applied three times. Let's make that precise.
Past Knowledge
Isosceles Triangle Theorem (5.1.4). Angle Sum (5.1.2). Classification (5.1.1).
Today's Goal
Prove and apply the equivalence of equilateral and equiangular triangles.
Future Success
Triangle congruence proofs (5.2), special right triangles, and coordinate proofs.
Key Concepts
Corollary 1: Equilateral → Equiangular
If a triangle is equilateral, then it is equiangular (all angles = 60°).
Why: Apply the Base Angles Theorem to each pair of congruent sides. Since , we get . By the Angle Sum Theorem, each = .
Corollary 2: Equiangular → Equilateral
If a triangle is equiangular, then it is equilateral (all sides congruent).
Why: Apply the Converse of the Base Angles Theorem to each pair of congruent angles. Since , we get .
Biconditional (If and Only If)
A triangle is equilateral if and only if it is equiangular. These are not separate types — they are two names for the same triangle.
Properties at a Glance
- All three sides are congruent.
- All three angles are .
- It is always acute (60° < 90°).
- It has 3 lines of symmetry and order-3 rotational symmetry (120°).
- It is a special case of an isosceles triangle (satisfies it three times over).
Worked Examples
Finding Side Length from Perimeter
An equilateral triangle has a perimeter of 42 cm. Find the side length and each angle.
All three sides are equal, so:
Since equilateral ⟹ equiangular, each angle = 60°.
Each side = 14 cm, each angle = 60°.
Algebraic Equilateral Condition
A triangle has sides , , and . Find the value of that makes it equilateral, and verify.
Set the first two sides equal:
Verify all three sides:
✓
All sides = 14, so the triangle is equilateral and therefore equiangular (all 60°).
. All sides = 14, all angles = 60°.
Coordinate Proof — Is It Equilateral?
Prove that with , , is equilateral.
All three sides have length 6.
, so is equilateral (and equiangular with 60° angles). ∎
Common Pitfalls
Treating Equilateral and Equiangular as Different Types
They are the same triangle described two ways. If you know one property, you automatically know the other. Don't prove them separately on a test — just cite the biconditional.
Setting Only Two Sides Equal
When solving for to make a triangle equilateral, you must verify that all three sides are equal at your computed value, not just two. Setting two equal gives isosceles — you need the third to match as well.
Assuming Regular Polygon Properties
The equilateral ⟺ equiangular equivalence holds only for triangles. A rhombus is equilateral but not equiangular. A rectangle is equiangular but not equilateral. For other polygons, these are independent properties.
Real-Life Applications
Structural Engineering — The Strongest Shape
The equilateral triangle distributes force equally among all three sides and angles, making it the most rigid 2D shape. This is why it appears in trusses, bridges, and geodesic domes — Buckminster Fuller's dome at Expo '67 is made entirely of equilateral triangles.
Tessellations & Tiling
Equilateral triangles are one of only three regular polygons that can tile the plane perfectly (along with squares and hexagons). The interior angle of 60° divides evenly into 360° (six triangles meet at each vertex), creating seamless patterns used in flooring, mosaics, and game boards.
Practice Quiz
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