Pentagon Calculator
Calculate area, perimeter, and diagonal with golden ratio insights
Input Values
Results
Step-by-Step Solution
φ = (1 + √5) / 2
φ = (1 + 2.236068) / 2
φ ≈ 1.618034
A = (s² × √(25 + 10√5)) / 4
A = (10² × √(25 + 10√5)) / 4
A ≈ 0.0000
d = s × φ
d = 10 × 1.618034
d ≈ 0.0000
P = 5 × s
P = 5 × 0.0000
P = 0.0000
Understanding Pentagons
What is a Pentagon?
A pentagon is a five-sided polygon. A regular pentagon has all five sides of equal length and all five interior angles equal to 108°. The regular pentagon is famous for its connection to the golden ratio and appears in nature, architecture, and the U.S. Pentagon building.
The Golden Ratio Connection
The regular pentagon has a special relationship with the golden ratio (φ ≈ 1.618). The ratio of a diagonal to a side equals φ. This means d = s × φ, where d is the diagonal and s is the side length. Additionally, the ratio of consecutive diagonal segments also equals φ, creating a geometric sequence.
Golden Ratio: φ = (1 + √5) / 2 ≈ 1.618033988...
Diagonal Formula: d = s × φ
Side from Diagonal: s = d / φ = d × φ⁻¹
Pentagon Formulas
Area: A = (s² × √(25 + 10√5)) / 4 ≈ 1.720477 × s²
Perimeter: P = 5s
Diagonal: d = s × φ ≈ 1.618s
Apothem: a = s / (2 × tan(36°)) ≈ 0.688s
Radius: r = s / (2 × sin(36°)) ≈ 0.851s
Interior Angle: 108°
Exterior Angle: 72°
Pentagon in Nature and Design
- Flowers like morning glories and okra have pentagonal symmetry
- Starfish and sea urchins exhibit five-fold radial symmetry
- The U.S. Pentagon building is a famous architectural example
- Pentagons appear in Islamic geometric art and tilings
- The pentagram (five-pointed star) contains multiple golden ratio relationships
Frequently Asked Questions
Why is the golden ratio in a pentagon?
The golden ratio appears naturally in a regular pentagon because of its geometric properties. When you draw all five diagonals, they form a pentagram, and the ratio of any diagonal to a side equals φ. This occurs because of the specific 72° and 36° angles in the pentagon's construction.
Can pentagons tessellate?
Regular pentagons cannot tessellate (tile a plane without gaps) because their 108° interior angles don't divide evenly into 360°. However, certain irregular pentagons can tessellate, and 15 distinct types of convex pentagons that tessellate have been discovered.
How do you construct a regular pentagon?
A regular pentagon can be constructed with compass and straightedge. One method involves drawing a circle, then using the golden ratio to find the side length. Another method uses the fact that cos(72°) = (√5 - 1)/4, which can be constructed geometrically.
What is the area formula derivation?
The area can be derived by dividing the pentagon into 5 congruent triangles from the center. Each triangle has base s and height equal to the apothem. Using trigonometry, a = s/(2tan(36°)), and the total area becomes A = (5/2) × s × a, which simplifies to A ≈ 1.720477s².
How many diagonals does a pentagon have?
A pentagon has 5 diagonals. The formula for the number of diagonals in any polygon is n(n-3)/2, where n is the number of sides. For a pentagon: 5(5-3)/2 = 5(2)/2 = 5. These 5 diagonals intersect to form a smaller pentagon inside.
What is a pentagram?
A pentagram is a five-pointed star formed by drawing all five diagonals of a regular pentagon. It's also called a pentacle and has been used as a symbol in various cultures. The pentagram itself contains many instances of the golden ratio in its line segment ratios.
Why is the interior angle 108°?
Using the formula for interior angles: (n-2) × 180° / n, we get (5-2) × 180° / 5 = 3 × 180° / 5 = 540° / 5 = 108°. This means the sum of all interior angles is 540°, with each angle being 108° in a regular pentagon.
What is five-fold symmetry?
Five-fold symmetry means an object looks the same after rotation by 72° (360°/5). Regular pentagons have this symmetry, as do many biological organisms like starfish. Interestingly, five-fold symmetry is forbidden in crystallography but appears in quasicrystals.