Pentagon Calculator
How It Works
A regular pentagon has five equal sides and five equal angles of 108 degrees each. Its area formula involves the square root expression: A = (sqrt(5(5+2sqrt(5)))/4) x s^2, which simplifies to approximately 1.72 x s^2.
The diagonal of a regular pentagon is related to the side length by the golden ratio (phi = 1.618...). Specifically, diagonal = s x phi. This mathematical relationship connects pentagons to the Fibonacci sequence and golden spiral, making them important in both mathematics and art.
Regular pentagons cannot tessellate (tile a plane without gaps), unlike triangles, squares, and hexagons. However, certain irregular pentagons can tessellate, and this was an active area of mathematical research until the complete classification in 2017.
Frequently Asked Questions
What is the area formula for a regular pentagon?
Area = (√(5(5+2√5))/4) × s² ≈ 1.7205 × s². For a pentagon with side 10, the area is approximately 172.05 square units.
How many diagonals does a pentagon have?
A regular pentagon has exactly 5 diagonals. Each diagonal has the same length: s × φ, where φ (phi) is the golden ratio ≈ 1.618. The diagonals form a smaller regular pentagon inside.
What is the connection between pentagons and the golden ratio?
The ratio of a pentagon diagonal to its side equals the golden ratio φ ≈ 1.618. The diagonals of a regular pentagon intersect to form a pentagram, and each intersection point divides the diagonal in the golden ratio.
What is the interior angle of a regular pentagon?
Each interior angle of a regular pentagon is 108 degrees. The sum of all interior angles is 540 degrees, calculated as (5-2) × 180°.