Golden Kite

"... the algorithmically constructive and creative definition ... models new objects out of familiar ones." - Hans Freudenthal (1973:458) in Mathematics as an Educational Task.

The concept of a 'golden rectangle' - a rectangle with its sides in the the golden ratio φ = (√5 + 1)/2 - can be extended to that of a 'golden kite' in different ways as shown below.

Definition 1: A golden kite is composed of a 'golden triangle' - which is an isosceles triangle with angles 36°, 72°, 72° - with an equilateral triangle constructed on its shorter side.
a) As shown in the dynamic sketch below, it immediately follows from the properties of a golden triangle that DA/BD = DA/DC = φ. However, the diagonals are not in the golden ratio.
b) Though one can also obtain a concave golden kite according to this definition as can be seen by clicking on the 'Show Concave case' button, we shall restrict our investigation to only convex cases.

Challenge 1: Prove that the sides of the golden triangle as well as the kite defined above are in the golden ratio; i.e. that DA/BD = DA/DC = φ (as experimentally shown by the measurements in the sketch below).

Definition 2: A golden kite ABCD is a kite with angles A, C and B = D in a geometric progression of φ, i.e. if ∠A = x, then ∠C = φx and B = D = φ²x. Click on the 'Link to Construction 2' button to view.
a) Note that none of the sides or the diagonals in this case are in the golden ratio. However, the ratio of diagonals are closer to φ than those of the golden kite for definition 1; hence it appears a little 'fatter'.

Challenge 2: Determine (calculate) the angles at A and D of a golden kite ABCD according to definition 2. Check your answers by clicking on the 'Show Angles at A and D' button.

Some Golden Kite Constructions

Definition 3: A golden kite ABCD is a kite with its sides and diagonals in the golden ratio φ. Click on the 'Link to Construction 3' button above to view.
a) Note that compared to the preceding case, ∠B is larger and ∠A is smaller; which explains why it appears a little 'fatter'.

Challenge 3 (medium-hard): Determine (calculate) the angles at A and B of a golden kite ABCD according to definition 3. Check your answers by clicking on the 'Show Angles at A and B' button.

Definition 4: A golden kite ABCD can be obtained from a 'golden isosceles trapezium - Type 3 (Def. 2)' by constructing tangents, at its vertices, to its circumcircle. Click on the 'Link to Construction 4' button above to view.
a) Note that according to this definition, a golden kite has three equal angles of 108° and is therefore a special 'triangular kite'.

Challenge 4
Prove that if a kite ABCD is constructed according to definition 4, then it has three equal angles of 108°. With reference to the sketch, also prove that a) CBD is a golden triangle, and b) the tangential points K and N are the respective midpoints of AB and AD.

Definition 5: Penrose kite & dart: A golden kite is a Penrose kite or Penrose dart. They can both be obtained from a rhombus with angles of 72° and 108°, by dividing the long diagonal of the rhombus in the ratio φ. Click on the 'Link to Penrose Kite & Dart' button above to view.
a) Note that from the above construction, the symmetrical diagonal of the Penrose kite is in the ratio φ to the symmetrical diagonal of the Penrose dart.

Challenge 5
a) Prove that both the Penrose kite and dart have their sides in the golden ratio φ.
b) Can you create a tiling with a penrose kite or dart together with a 'golden isosceles trapezium - Type 3 (Def. 2)'? Investigate before checking & comparing your findings with De Villiers (2018).
c) Can you produce an aperiodic tiling with the Penrose kite and dart? In other words, can you create a tiling with these tiles that has no translation symmetry?

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Critical Comparison
Critically compare the possible definitions of a 'golden kite' above, judging them not only by 'appearance' or 'shape', but also by their mathematical properties & the ease by which one can construct an example or deduce other additional properties (not contained in the definition).
a) Which one of these do you most prefer? Why?

Other Golden Quadrilaterals
To view, interact with & explore other golden quadrilaterals CLICK HERE.

Constructive Defining
Read more about constructive (and descriptive) defining and classifying in mathematics in De Villiers (2009).

References
De Villiers, M. (1994, 1996, 2009). Defining and Classifying (Chapter 2), pp. 13-28, from Some Adventures in Euclidean Geometry. (Free to download). Morrisville, NC: Lulu Publishers, Dynamic Mathematics Learning.
De Villiers, M. (2017). An Example of Constructive Defining: From a Golden Rectangle to Golden Quadrilaterals. At Right Angles, Vol. 6, No. 1, pp. 64-69; Vol. 6, No. 2, pp. 74-81.
De Villiers, M. (2018). Tiling with a Trilateral Trapezium and Penrose Tiles. Learning & Teaching Mathematics, No. 25, pp. 6-10.

External Links
Golden rectangle (Wikipedia)
Golden ratio (Wikipedia)
Golden triangle (Wikipedia)
Kite (geometry) (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Created by Michael de Villiers, 20 February 2022 with WebSketchpad; updated 22 July 2025; 24 April 2026.

Golden Kite

.sketch_canvas { border: medium solid lightgray; display: inline-block; } Some Golden Kite Constructions

Some Golden Kite Constructions