cyberangles blog

Largest Square Inscribed in a Hexagon Inscribed in an Equilateral Triangle

Geometric constructions involving nested polygons offer fascinating challenges that blend trigonometry, algebra, and spatial reasoning. This blog explores a specific problem: finding the largest possible square inscribed within a regular hexagon, which itself is inscribed within an equilateral triangle. This sequence—equilateral triangle → regular hexagon → square—has practical applications in areas like material optimization, architectural design, and computational geometry. We'll derive the solution step by step, provide practical examples, and outline best practices for solving similar problems.

2026-06

Table of Contents#

  1. Problem Statement and Key Assumptions
  2. Inscribing a Regular Hexagon in an Equilateral Triangle
    • Geometric Construction
    • Side Length Derivation
  3. Inscribing the Largest Square in the Regular Hexagon
    • Horizontal Alignment Approach
    • Side Length Derivation
    • Alternative Orientations
  4. Combined Solution
    • Final Formula Derivation
  5. Examples and Verification
  6. Best Practices for Geometric Inscription Problems
  7. Conclusion
  8. References

1. Problem Statement and Key Assumptions#

Problem:
Given an equilateral triangle with side length (S), inscribe a regular hexagon within it. Then, inscribe the largest possible square within this hexagon. Derive the side length of this square in terms of (S).

Assumptions:

  • The equilateral triangle is the outer shape.
  • The hexagon is regular and inscribed such that three of its sides lie on the three sides of the triangle.
  • The square shares its center with the hexagon and is rotated to maximize its area.

Key Insights:

  • Symmetry is crucial—all nested shapes share rotational and reflectional symmetry.
  • Coordinate geometry simplifies derivation by positioning shapes in a Cartesian plane.

2. Inscribing a Regular Hexagon in an Equilateral Triangle#

Geometric Construction#

  1. Divide each side of the equilateral triangle into three equal segments of length (S/3).
  2. From each vertex, mark the points at (S/3) and (2S/3) along the sides.
  3. Connect adjacent points:
    • On side (AB), let (P_1) be at (S/3) from (A), (P_2) at (2S/3) from (A).
    • Similarly, define points on sides (BC) and (CA).
  4. Form the hexagon: Connect (P_2 \to R_1 \to Q_2 \to P_1 \to R_2 \to Q_1 \to P_2) (see diagram below).

Why is this hexagon regular?
Connecting these points removes three small equilateral triangles (side (S/3)) from the corners. The remaining hexagon has equal sides and internal angles, making it regular.

Side Length Derivation#

Each side of the hexagon corresponds to the segment between points like (P_1) on (AB) and (R_2) on (AC). The distance is:

  • Vector difference: (P_1 - R_2 = \text{segment of length } S/3).
  • Thus, hexagon side length (A = \frac{S}{3}).

Hexagon in Triangle Diagram
Figure: Regular hexagon (blue) inscribed in equilateral triangle (black) via corner truncation.

3. Inscribing the Largest Square in the Regular Hexagon#

Horizontal Alignment Approach#

Place the hexagon in a "flat-top" orientation (sides parallel to the x-axis) with center ((0,0)):

  • Vertices at:
    • (H_1: (A, 0))
    • (H_2: (A/2, A \sqrt{3}/2))
    • (H_3: (-A/2, A \sqrt{3}/2))
    • (H_4: (-A, 0))
    • (H_5: (-A/2, -A \sqrt{3}/2))
    • (H_6: (A/2, -A \sqrt{3}/2))

To maximize the square:

  • Center it at ((0,0)) for symmetry.
  • Align sides horizontally/vertically.
  • Define square by:
    • Top edge: (y = k)
    • Bottom edge: (y = -k)
    • Side length: (s = 2k) (since centered).

Constraints#

  1. Top edge at (y = k) must fit within hexagon width:

    • Hexagon width at (y = k): (2\left(A - \frac{|k|}{\sqrt{3}}\right)).
    • Constraint: (s \leq 2\left(A - \frac{k}{\sqrt{3}}\right)).
    • Substitute (k = s/2): [ s \leq 2\left(A - \frac{s/2}{\sqrt{3}}\right) \implies s \leq \frac{2A \sqrt{3}}{\sqrt{3} + 1} ]
    • Simplify: [ s \leq A(3 - \sqrt{3}) \quad (\text{since } \frac{2\sqrt{3}}{\sqrt{3}+1} = 3 - \sqrt{3}) ]

  2. Vertical edges (left at (x = -s/2), right at (x = s/2)):

    • At any (y) in ([-s/2, s/2]), ensure: [ -\frac{s}{2} \geq \frac{y}{\sqrt{3}} - A \quad \text{and} \quad \frac{s}{2} \leq A - \frac{|y|}{\sqrt{3}}. ]
    • Maximum constraint (at (y = s/2)) matches the top-edge constraint.

Thus, maximum square side: [ s_{\text{max}} = A(3 - \sqrt{3}) ]

Alternative Orientations#

Rotating the square (e.g., by 30°) yields smaller squares:

  • Rotated solutions: (A(\sqrt{3} - 1)) or (A(\sqrt{6} - \sqrt{2})).
  • Horizontal alignment gives the largest: [ A(3 - \sqrt{3}) \approx 1.268A > A(\sqrt{3}-1) \approx 0.732A ]

4. Combined Solution#

Substitute (A = S/3) into (s_{\text{max}}): [ \boxed{s = \dfrac{S(3 - \sqrt{3})}{3}} ]

This is the side length of the largest square in terms of (S).

5. Examples and Verification#

Example 1: (S = 3)#

  • Hexagon side: (A = 3/3 = 1).
  • Square side: (s = (3 - \sqrt{3}) \approx 3 - 1.732 = 1.268).
  • Verification:
    • Hexagon vertices: ((0, \sqrt{3}/2)), ((0.5, 0)), etc. (skipped for brevity).
    • Square vertices: ((\pm 0.634, 0.634)), ((0.634, -0.634)), etc.
    • Distance from center to edge: (0.634^2 + 0.634^2 \approx 0.804 < \text{hexagon apothem } (\sqrt{3}/2 \approx 0.866)). ✓

Example 2: (S = 6)#

  • Hexagon side: (A = 6/3 = 2).
  • Square side: (s = 2(3 - \sqrt{3}) \approx 2 \times 1.268 = 2.536).

6. Best Practices for Geometric Inscription Problems#

  1. Leverage Symmetry: Align shapes to share centers/axes for simpler math.
  2. Coordinate Geometry: Plot shapes to derive constraints and verify results.
  3. Optimization Workflow:
    • Derive inner shape parameters first (e.g., hexagon side (A = S/3)).
    • Maximize subsequent shapes (e.g., iterate square orientations).
    • Validate with vertex checks or distance formulas.
  4. Practical Applications:
    • Material Design: Minimize waste by nesting shapes optimally.
    • Architecture: Structural layouts for windows, tiles, or supports.
    • Algorithms: Computational methods for packing/inscription (e.g., rotational checks).

7. Conclusion#

We derived the largest square inscribed in a hexagon inscribed in an equilateral triangle: [ s = \frac{S(3 - \sqrt{3})}{3} ] Key steps included hexagon truncation, horizontal alignment for the square, and constraint optimization. This approach demonstrates how geometric properties and coordinate systems solve complex nesting problems efficiently. The methodology applies to similar challenges with polygons or rotational optimizations.

8. References#

  1. Weisstein, E. W. "Regular Hexagon." MathWorld.
  2. Alsina, C., & Nelsen, R. (2010). Charming Proofs: A Journey Into Elegant Mathematics. MAA.
  3. Stack Exchange (Mathematics). "Largest Square in a Hexagon."
  4. Honsberger, R. (1976). Mathematical Gems II. MAA.