What is a Golden Triangle?

In everyday life, we see a lot of different shapes like a study table is of a rectangle or square shape, water bottle is cylindrical, the globe is a sphere, the watch is circular, etc. To study these shapes, sizes, angles a branch of mathematics is required, and that branch is geometry. 

Golden Triangle

It is not the triangle having golden color as the name suggests. Before diving into it let’s understand the Golden ratio first.

Golden ratio

It is the number that is used to denote the ratio of distances in geometry. It is denoted by ‘phi’ or φ. Two dimensions are representing a golden ratio, if the ratio of these 2 dimensions is equal to the ratio of the sum of these dimensions by the largest of these two dimensions. In other words,

(a + b)/a = a/b = φ, where a > b > 0

Talking about φ,

φ is an irrational number and it is a solution of the quadratic equation x² − x − 1 = 0, 

Value of φ = 5√2 + 1 = 1.6180339887… = approx 1.62

Now, let’s understand the concept of the golden triangle.

The golden triangle is an isosceles triangle in which the ratio of a and b from the figure above, in other words, the ratio of the hypotenuse and base is equal to the golden ratio.

a/b = phi or φ.

It is also known as the sublime triangle.

So, the vertex angle is equal to,

θ = sin^-1(b ⁄ (2 × a)) 

θ = 2 × sin^-1(1  ⁄  (2 × φ))

θ =1  ⁄  (5 × π) = 36° 

So, the height (let h) and base (let b) of this triangle is related as,

4 × h² = b² × (5 + 2√5) 

Finding two quantities with a property of golden ratio

Let’s go step by step-

Step-1: First create an isosceles triangle with interior angles of 72°, 36° and 72°.

Step-2  Bisect one of the 72° angle.

Step-3 Now solve value of angle α.

In ΔABD, 

36°+ 36° + α = 180°

72° + α = 180°

α = 108°

Value of α = 108°

Step-4 Now solve value of angle β.

In ΔACD,

36°+ 72° + β = 180°

108° + β = 180°

β = 72°

Value of β = 72°

Step-5 Now ΔABC and ΔDAC are similar triangles because their angles are same. 

Therefore by similar triangles property,

BC  ⁄  AC = AD  ⁄  DC ⇢ eq (1)

But triangle ABD is also isosceles, Therefore BD = AD, and ΔADC is also isosceles so, AD = AC, 

Thus, BD = AD = AC ⇢ eq (2)

From eq (1) and eq (2):

BC ⁄ BD = BD ⁄ DC

Point D divides line BC into a golden ratio, Let’s Compare

BC ⁄ BD = BD ⁄ DC is same as (a + b) ⁄ a = a ⁄ b, i.e, golden ratio.

Golden gnomon

In the above figure, two types of triangles are there, 

1. ΔABD – angles are in 1:1:3 proportion

2. ΔADC – angles are in 2:2:1 proportion.

Now, the triangle having angles in 2:2:1 proportion is called a golden triangle. On the other hand, the triangle having angles in 1:1:3 proportion is called a Golden gnomon.

Golden gnomon – It is an obtuse isosceles triangle having a ratio of the length of the equal (shorter) sides to the length of the third side is the reciprocal of the golden ratio. 

Relation between golden triangles and the Pythagoras Pentagon 

In the above figure, the pentagon is filled with golden triangles (green color) and golden gnomon (grey color). Let’s find the ratio of the inner pentagon (dark green color in the below figure) to the area of the entire figure.

a ⁄ b = ϕ (golden ratio) and b ⁄ c = ϕ, so by this a ⁄ c = ϕ².

As they both are similar figures, 

So it can be concluded,

Area of inner pentagon (dark green color) = 1 ⁄  ϕ⁴ × (entire pentagon area).

Applications of Golden Triangle

Now, after understanding the golden triangle one must be thinking, about how it is used in real life, so let’s deep dive into its applications.

1. In Architecture: Used in determining the dimensions of the layouts.
2. Logarithmic Spiral: Golden triangles are used to form some points of the logarithmic spiral.
3. It has many other applications in paintings, presentations, photographs.

Sample Problems

Question 1: Check if triangle ABC is golden triangle or not where angle A = 68°, B = 41°, C = 71°.

Solution:

No.

Explanation:

By observing carefully, 

Triangle ABC is not an isosceles triangle as no two angles are equal. So, it can’t be a golden triangle as for being golden triangle mandatory condition is to being isosceles first.

Question 2: Check if triangle ABC is golden triangle or not where side AB = 208 cm, B = 203 cm, C = 145 cm.

Solution:

No.

Explanation:

By observing carefully,

Triangle ABC is not an isosceles triangle as no two sides are equal. So, it can’t be a golden triangle as for being golden triangle mandatory condition is to being isosceles first.

Question 3: Check if two sides of triangle are in golden ratio or not AB = 61.77, BC = 38.22.

Solution:

Yes.

Explanation:

1.  AB ⁄ BC = 61.77  ⁄  38.22 = 1.61 approx

2. (AB + BC) ⁄ AB =  99.99 ⁄  61.77 = 1.61 approx

As, both 1 and 2 are equal so they are in golden ratio.

Question 4: Check if two sides of triangle are in golden ratio or not AB = 0.618, BC = 1.333.

Solution:

No.

Explanation:

Largest quantity should be in denominator

1.  BC ⁄ AB = 1.333 ⁄ 0.618 = 2.15 approx

2. (AB + BC) ⁄ BC =  1.951 ⁄ 1.333 = 1.46 approx

As, both 1 and 2 are not equal so they are not in golden ratio.

Question 5: Check if triangle ABC is golden triangle or not where angle A = 72°, B = 72°, C = 36°.

Solution:

Yes.

Explanation:

By observing carefully,

Triangle ABC is an isosceles triangle as two angles are equal. And ratio of angles is 2:2:1 which is standard property for being a golden triangle.

Question 6: Check if triangle ABC is golden triangle or not where angle A = 76°, B = 72°, C = 34°.

Solution:

No.

Explanation:

By observing carefully,

Triangle ABC is not even a valid triangle as sum of angles A + B + C = 76 + 72 + 34 = 182°. But the sum of angles of triangle should be 180°.