Geometry is full of hidden treasures, and the 36°–36°–108° triangle is undoubtedly one of them. Although it is simply an isosceles triangle with two equal angles of 36°, it possesses remarkable properties that connect it directly to the Golden Ratio, the regular pentagon, and the beautiful geometry of the pentagram. It is a Natural and Permanent Resident of the Regular Pentagon!
Suppose the two equal sides each have length 1, and let the base have length b.
In a 36°–36°–108° triangle, the ratio of the base to either equal side is exactly the Golden Ratio.
If the equal sides are 1 unit long, the base measures approximately (√5+1)/2 or 1.618 units.
Take any regular pentagon and draw one of its diagonals. The triangle formed by that diagonal and the two adjacent sides is a 36°–36°–108° triangle.
This explains why the Golden Ratio appears so naturally in pentagonal geometry. In fact, the diagonal of a regular pentagon is exactly (√5+1)/2 or Golden Ratio times as long as one of its sides.
If we draw the altitude from the 108° vertex to the base.
This divides the triangle into two congruent right triangles, each having angles
18°, 36°, and 90°.These right triangles are themselves rich in elegant identities and often appear in olympiad problems and geometric constructions.
The 36°–36°–108° triangle is a wonderful reminder that extraordinary mathematics can arise from an ordinary looking figure. Its intimate relationship with the Golden Ratio, regular pentagons, and classical geometry makes it a favourite among puzzle enthusiasts and mathematicians alike.
Once you learn to recognize this triangle, you'll begin spotting it in pentagrams, geometric proofs, and competition problems. And you'll discover that it often unlocks elegant solutions with very little computation.
The various features of 36°–36°–108° Triangle are presented in a poster, which can be sent to students or maths lovers alike.
