Pentagon Calculator
How a Regular Pentagon Works
A regular pentagon is a five-sided polygon with all sides of equal length and all interior angles equal to 108 degrees. It is one of the fundamental shapes in Euclidean geometry, studied since the time of the ancient Greeks. According to Wolfram MathWorld, the regular pentagon is unique among polygons for its deep connection to the golden ratio (phi = 1.6180339...), which appears in the relationship between its diagonal and side length. The ancient Pythagoreans used the pentagram (a five-pointed star inscribed in a pentagon) as their secret symbol because of these mathematical properties.
Pentagons appear throughout nature and human design. The Pentagon building in Arlington, Virginia, is the world's largest office building by floor area at 6.5 million square feet. In nature, many flowers have five petals (roses, buttercups, hibiscus), starfish have five arms, and the cross-section of an apple reveals a pentagonal seed arrangement. This calculator computes the area, perimeter, diagonal, and apothem of a regular pentagon from the side length.
The Pentagon Formulas
All measurements of a regular pentagon derive from a single value: the side length s. The key formulas are:
- Area: A = (sqrt(5(5 + 2*sqrt(5))) / 4) * s^2, which simplifies to approximately 1.7205 * s^2
- Perimeter: P = 5 * s
- Diagonal: d = s * phi = s * (1 + sqrt(5)) / 2, approximately 1.6180 * s
- Apothem: a = s / (2 * tan(pi/5)), approximately 0.6882 * s
- Circumradius: R = s / (2 * sin(pi/5)), approximately 0.8507 * s
Worked example: For a pentagon with side length s = 10: Area = 1.7205 * 100 = 172.05 sq units, Perimeter = 50 units, Diagonal = 16.18 units, Apothem = 6.88 units.
Key Terms You Should Know
- Regular Pentagon -- A five-sided polygon where all sides are equal and all interior angles are 108 degrees. An irregular pentagon has unequal sides or angles.
- Golden Ratio (phi) -- The irrational number (1 + sqrt(5)) / 2, approximately 1.6180. It equals the ratio of any pentagon diagonal to its side length and appears throughout mathematics, art, and nature.
- Apothem -- The perpendicular distance from the center of the pentagon to the midpoint of any side. It forms a right angle with the side and is used in the alternative area formula A = (1/2) * P * a.
- Pentagram -- The five-pointed star shape formed by drawing all five diagonals of a regular pentagon. Each intersection point divides the diagonal in the golden ratio.
- Circumradius -- The distance from the center of the pentagon to any vertex. It is the radius of the circumscribed circle that passes through all five vertices.
Regular Polygon Comparison Table
The table below compares properties of regular polygons with a side length of 10, showing how the pentagon relates to other common shapes. As the number of sides increases, the polygon approaches a circle.
| Polygon | Sides | Interior Angle | Area (s=10) | Perimeter | Tessellates? |
|---|---|---|---|---|---|
| Equilateral Triangle | 3 | 60 | 43.30 | 30 | Yes |
| Square | 4 | 90 | 100.00 | 40 | Yes |
| Regular Pentagon | 5 | 108 | 172.05 | 50 | No |
| Regular Hexagon | 6 | 120 | 259.81 | 60 | Yes |
| Regular Octagon | 8 | 135 | 482.84 | 80 | No |
| Regular Decagon | 10 | 144 | 769.42 | 100 | No |
Practical Examples
Example 1: Garden Patio. You want to build a pentagonal patio with each side measuring 3 meters. Area = 1.7205 * 3^2 = 1.7205 * 9 = 15.48 square meters. Perimeter = 5 * 3 = 15 meters. You would need approximately 15.5 square meters of paving material (add 10% for waste) and 15 meters of border edging.
Example 2: The Pentagon Building. The Pentagon in Arlington, Virginia has outer walls of approximately 281 meters (921 feet) per side. Its area = 1.7205 * 281^2 = approximately 135,846 square meters (1.46 million square feet of footprint). The actual building has five concentric pentagonal rings with a 5-acre central courtyard.
Example 3: Decorative Tile. A craftsman is making pentagonal ceramic tiles with 8 cm sides. Each tile has area = 1.7205 * 64 = 110.1 square cm. To cover a 1 square meter (10,000 sq cm) surface, he would need approximately 91 tiles (accounting for the gaps since regular pentagons cannot tile a surface without leaving spaces -- he would need filler shapes). Using our area calculator can help with the overall surface area estimate.
Tips for Working with Pentagons
- Drawing a regular pentagon. Use a compass and straightedge. Draw a circle, mark the center, and divide the circle into five equal arcs of 72 degrees each. Connect the division points to form the pentagon.
- Approximate check. A regular pentagon's area is approximately 1.72 times the square of its side length. If your calculated area differs significantly from 1.72 * s^2, double-check your measurements.
- Using the apothem for area. If you know the apothem (center-to-side distance) rather than the side length, use A = (1/2) * perimeter * apothem = (1/2) * 5s * a. This is often more convenient when measuring real objects.
- Converting between measurements. If you only know the diagonal, calculate the side as s = d / phi = d / 1.618. If you know the circumradius (center-to-vertex), the side is s = 2R * sin(pi/5) = 1.1756 * R.
- Remember tessellation limits. Regular pentagons cannot tile a flat surface without gaps. If you need pentagonal-themed tiling, use a combination of pentagons and other shapes, or use the 15 known types of irregular convex pentagons that do tessellate.
Frequently Asked Questions
What is the area formula for a regular pentagon?
The area of a regular pentagon is calculated using A = (sqrt(5(5 + 2*sqrt(5))) / 4) * s^2, which simplifies to approximately 1.7205 * s^2. For a pentagon with side length 10, the area is approximately 172.05 square units. This formula can also be expressed as A = (5 * s^2) / (4 * tan(pi/5)), which is derived from dividing the pentagon into five congruent isosceles triangles meeting at the center.
How many diagonals does a pentagon have?
A regular pentagon has exactly 5 diagonals, calculated using the formula n(n-3)/2 = 5(5-3)/2 = 5. Each diagonal has the same length: d = s * phi, where phi is the golden ratio approximately 1.6180. The five diagonals intersect inside the pentagon to form a smaller regular pentagon, and the resulting star shape is called a pentagram. This self-similar nesting of pentagons within pentagrams can continue infinitely.
What is the connection between pentagons and the golden ratio?
The regular pentagon is deeply connected to the golden ratio phi = (1 + sqrt(5))/2, approximately 1.6180. The ratio of any diagonal to any side of a regular pentagon equals phi exactly. When diagonals intersect, each intersection point divides the diagonal in the golden ratio. The ancient Greeks discovered this relationship, and the Pythagoreans used the pentagram (five-pointed star formed by pentagon diagonals) as their secret symbol precisely because of this mathematical property.
What is the interior angle of a regular pentagon?
Each interior angle of a regular pentagon is exactly 108 degrees. The sum of all interior angles is 540 degrees, calculated using the polygon angle sum formula (n-2) * 180 = (5-2) * 180 = 540 degrees. Each exterior angle is 72 degrees (180 - 108), and the exterior angles sum to 360 degrees as with any convex polygon. The 108-degree angle is significant because 108 = 3 * 36, and 36 degrees is the base angle of the golden gnomon triangle.
Can regular pentagons tile a flat surface?
Regular pentagons cannot tessellate (tile a flat plane without gaps or overlaps) because the interior angle of 108 degrees does not divide evenly into 360 degrees. Only three regular polygons tessellate on their own: equilateral triangles (60 degrees), squares (90 degrees), and regular hexagons (120 degrees). However, certain irregular convex pentagons can tessellate. Mathematicians discovered 15 types of convex pentagonal tilings between 1918 and 2015, and a proof that the list is complete was published in 2017.
What is the apothem of a regular pentagon?
The apothem is the perpendicular distance from the center of a regular pentagon to the midpoint of any side. It is calculated as a = s / (2 * tan(pi/5)), which equals approximately 0.6882 * s. The apothem is useful because the area of any regular polygon can be calculated as A = (1/2) * perimeter * apothem. For a pentagon with side 10, the apothem is approximately 6.882 and the area is (1/2) * 50 * 6.882 = 172.05 square units.