Calculate Tri-Equilateral Trapezoid

Calculator and formulas for trapezoids with three equal sides

Tri-Equilateral Trapezoid Calculator

Special trapezoid form

In a tri-equilateral trapezoid three sides are equal in length: the two legs and one of the parallel sides. The fourth side has a different length.

Enter side lengths
The side that differs in length
The three equal sides
Calculation Results
Diagonal d:
Height h:
Middle width m:
Area A:
Perimeter P:
Angle α:
Angle β:
Circumradius rc:

Tri-Equilateral Trapezoid

Tri-Equilateral Trapezoid Circumcircle Trapezoid

Tri-equilateral trapezoid with three equal sides b and one unequal side a.
The lower diagram shows the circumcircle.

The tri-equilateral trapezoid

A tri-equilateral trapezoid is a special form of trapezoid:

  • Three equal sides: Both legs and one parallel side share the same length b
  • One unequal side: The other parallel side has length a ≠ b
  • Symmetry: Mirror-symmetric about the mid-perpendicular
  • Special properties: Combines regularity with asymmetry
  • Circumcircle: Has a unique circumcircle
  • Use: Common in architecture and design

Symmetry and special properties

The tri-equilateral trapezoid exhibits unique geometric properties:

Axis of symmetry
  • Mirror-symmetric about the mid-perpendicular
  • Divides into two congruent right triangles
  • Isosceles triangles at the ends
  • Symmetric angle distribution
Circumcircle property
  • All four vertices lie on a circle
  • Unique circumradius determined
  • Circle center on the symmetry axis
  • Chord properties of the sides

Mathematical particularities

The geometric properties lead to special mathematical relations:

Formula simplification
  • Simplified diagonal formula due to symmetry
  • Height calculation via Pythagoras possible
  • Direct relationship between a and b
  • Perimeter as simple sum: P = a + 3b
Angle relations
  • Base angles are complementary: α + β = 180°
  • Symmetric angles on equal sides
  • Arccos for angle computation
  • Trigonometric simplifications possible

Practical applications

Tri-equilateral trapezoids are used in various fields:

Architecture & Construction
  • Roof structures with special shapes
  • Window and door frame design
  • Bridge geometry and support structures
  • Modern facade design
Design & Art
  • Graphic design and logos
  • Jewelry designs and ornaments
  • Furniture design with geometric shapes
  • Contemporary art and sculptures
Engineering & Mechanical Design
  • Gear components and tooth profiles
  • Optical elements and prisms
  • Vehicle body parts
  • Tooling design and fixtures
Science & Research
  • Crystal structure analysis
  • Molecular geometry studies
  • Optics and wave theory
  • Mathematical modeling

Formulas for the tri-equilateral trapezoid

Diagonal d
\[d = \sqrt{b \cdot (a + b)}\]

Simplified formula due to symmetry

Height h
\[h = \frac{\sqrt{4b^2 - (a-b)^2}}{2}\]

From a right-angled subtriangle

Middle width m
\[m = \frac{a + b}{2}\]

Arithmetic mean of the parallel sides

Area A
\[A = \frac{(a + b) \cdot h}{2}\]

Standard trapezoid area formula

Perimeter P
\[P = a + 3b\]

Simplified: one side a + three sides b

Angle α
\[\alpha = \arccos\left(\frac{g^2 + b^2 - h^2}{2gb}\right)\]
with \(g = \frac{|a-b|}{2}\)

Via law of cosines in the subtriangle

Angle β
\[\beta = 180° - \alpha\]

Complementary angle to α

Circumradius rc
\[r_c = b \cdot \sqrt{\frac{b(a+b)}{4b^2-(a-b)^2}}\]

Radius of the unique circumcircle

Calculation Example

Given
Unequal side a = 3 Equal sides b = 5

Find: All properties of the tri-equilateral trapezoid

1. Calculate diagonal
\[d = \sqrt{b \cdot (a + b)}\] \[d = \sqrt{5 \cdot (3 + 5)} = \sqrt{5 \cdot 8} = \sqrt{40} \approx 6.32\]

Simplified diagonal formula

2. Calculate height
\[h = \frac{\sqrt{4b^2 - (a-b)^2}}{2}\] \[h = \frac{\sqrt{4 \cdot 25 - (3-5)^2}}{2} = \frac{\sqrt{100 - 4}}{2} = \frac{\sqrt{96}}{2} \approx 4.90\]

From a right-angled subtriangle

3. Area
\[A = \frac{(a + b) \cdot h}{2}\] \[A = \frac{(3 + 5) \cdot 4.90}{2} = \frac{8 \cdot 4.90}{2} = 19.60\]

Standard trapezoid area formula

4. Perimeter
\[P = a + 3b\] \[P = 3 + 3 \cdot 5 = 3 + 15 = 18\]

Simplified perimeter formula

5. Summary of all results
Diagonal d ≈ 6.32 Height h ≈ 4.90 Middle width m = 4.00 Area A = 19.60
Perimeter P = 18.00 Angle α ≈ 70.9° Angle β ≈ 109.1° Circumradius rc ≈ 3.24

Complete characterization of the tri-equilateral trapezoid

The tri-equilateral trapezoid in theory and practice

The tri-equilateral trapezoid is a fascinating geometric shape that combines regularity with asymmetry. With three equal sides and one differing side it provides a unique balance between symmetry and variation. This special property makes it both mathematically interesting and practically valuable.

Geometric characteristics

The special properties of the tri-equilateral trapezoid arise from its unique structure:

  • Three equal sides: Both legs and one parallel side have the same length b
  • One differing side: The other parallel side has a different length a
  • Axis symmetry: Symmetric about the mid-perpendicular of the parallel sides
  • Existence of circumcircle: Has a unique circumcircle through all four vertices
  • Special angle relations: Complementary base angles with trigonometric properties

Mathematical particularities

The mathematical properties lead to elegant formulas and relations:

Simplified calculations

Symmetry allows simplified formulas, especially for diagonal and perimeter. The equality of three sides reduces complexity significantly.

Trigonometric relations

Angle computation uses arccos, where the special side ratios lead to characteristic angles.

Circumcircle properties

As one of the few trapezoid types it has a circumcircle, leading to interesting geometric constructions.

Limit cases

For a = b it becomes a rhombus, for a = 0 it becomes an isosceles triangle - important special cases.

Constructive properties

The tri-equilateral trapezoid offers unique constructive advantages:

Stability and strength
  • Even load distribution through three equal sides
  • Reduced stress concentrations
  • Symmetric force transfer
  • High torsional stiffness
Manufacturing benefits
  • Only two different side lengths required
  • Simplified tooling production
  • Reduced material variety
  • Modular construction possible
Aesthetic qualities
  • Balanced proportions
  • Natural-looking asymmetry
  • Harmonic form language
  • Versatile design possibilities
Functional flexibility
  • Adjustable proportions via the a/b ratio
  • Scalable base form
  • Combinable with other shapes
  • Universally applicable

Modern applications and trends

In contemporary practice the tri-equilateral trapezoid gains relevance:

  • Parametric design: CAD systems use the clear mathematical relations for automated construction
  • Sustainable architecture: Optimized material use via standardized components
  • Digital fabrication: 3D printing and CNC benefit from simplified geometries
  • Biomimetics: Inspiration from natural forms with similar proportions
  • Smart materials: Adaptive structures exploit special mechanical properties
  • Modular construction: Prefabricated elements for efficient building
Summary

The tri-equilateral trapezoid represents a unique combination of mathematical elegance and practical applicability. Its special features — three equal sides, axis symmetry and the existence of a circumcircle — make it a valuable tool in geometry, construction and design. The simplified calculation formulas and versatile application possibilities demonstrate the potential of this often overlooked geometric shape.

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