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⬟ Pentagon Calculator
Calculate all properties of regular pentagons
📏 Pentagon Dimensions
📊 Results
Area
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Perimeter
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Diagonal
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Inradius (r)
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Circumradius (R)
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Interior Angle
108°
Exterior Angle
72°
Pentagon Visualization
📚 Understanding Regular Pentagons
What is a Regular Pentagon?
A regular pentagon is a five-sided polygon where all sides have equal length and all interior angles are equal (108°). It's one of the most recognizable geometric shapes, appearing in nature, architecture, and design. The pentagon has fascinating mathematical properties related to the golden ratio.
Pentagon Formulas
For a regular pentagon with side length a:
Area = (√(25 + 10√5) / 4) × a² ≈ 1.72048 × a²
Perimeter = 5 × a
Diagonal = φ × a ≈ 1.618 × a (where φ is the golden ratio)
Inradius (Apothem) = (a / 2) × √(5 + 2√5) ≈ 0.688 × a
Circumradius = (a / 2) × √(5 + √5) ≈ 0.851 × a
Pentagon Properties
- Number of Sides: 5 equal sides
- Number of Vertices: 5 vertices
- Number of Diagonals: 5 diagonals
- Interior Angle: 108° at each vertex
- Exterior Angle: 72° at each vertex
- Sum of Interior Angles: 540° (5 × 108°)
- Lines of Symmetry: 5 lines of symmetry
- Rotational Symmetry: Order 5 (72° rotations)
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 length equals the golden ratio. This makes the pentagon appear in many natural phenomena and artistic compositions that follow golden ratio proportions.
Real-World Applications
- Architecture: The Pentagon building in Washington D.C., various fortress designs
- Nature: Starfish, flowers like morning glories, cross-sections of okra and apples
- Design: Logos, badges, medals, and decorative patterns
- Military: Pentagon-shaped fortifications for optimal defense coverage
- Chemistry: Molecular structures like cyclopentane
- Sports: Home plate in baseball is a pentagon
How to Draw a Regular Pentagon
- Using a Compass: Draw a circle, mark 5 points at 72° intervals, connect the points
- Using Coordinates: Place vertices at angles 0°, 72°, 144°, 216°, 288° from center
- Folding Method: Create a pentagon by folding a strip of paper into a knot
- Construction: Use compass and straightedge following classical geometric construction
Frequently Asked Questions
What makes a pentagon "regular"?
A regular pentagon has all five sides of equal length and all five interior angles equal to 108°. If the sides or angles are unequal, it's an irregular pentagon. Regular pentagons have perfect symmetry with 5 lines of symmetry and rotational symmetry of order 5.
How is the pentagon related to the golden ratio?
In a regular pentagon, the ratio of a diagonal to a side length equals the golden ratio (φ ≈ 1.618). This means if you draw all the diagonals, they intersect in golden ratio proportions, creating a smaller pentagon inside. This relationship makes the pentagon aesthetically pleasing and appears frequently in nature.
Why is each interior angle 108°?
The sum of interior angles in any polygon is (n-2) × 180°, where n is the number of sides. For a pentagon: (5-2) × 180° = 540°. Since all angles are equal in a regular pentagon, each angle is 540° ÷ 5 = 108°. The exterior angles are 72° each (180° - 108°).
Can pentagons tessellate (tile a plane)?
Regular pentagons cannot tessellate by themselves because their 108° angles don't divide evenly into 360°. However, irregular pentagons can tessellate, and there are 15 known types of convex pentagons that can tile a plane. Regular pentagons can tessellate when combined with other shapes.
What is the apothem of a pentagon?
The apothem (also called inradius) is the distance from the center of the pentagon to the midpoint of any side. It's the radius of the largest circle that can fit inside the pentagon. For a regular pentagon with side length a, the apothem is approximately 0.688 × a.
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. These diagonals all intersect inside the pentagon and create a smaller pentagon in the center.
Where do we see pentagons in nature?
Pentagons appear frequently in nature: starfish have five-fold symmetry, many flowers have five petals (roses, apple blossoms, morning glories), cross-sections of okra and apples show pentagonal patterns, and some viruses have pentagonal symmetry. This five-fold symmetry is common in living organisms.
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