Calculate properties of regular polygons (n-gons).
Polygon Properties
Visual Representation
How to Use the Regular Polygon Calculator
This calculator helps you determine various geometric properties of any regular polygon (an n-sided polygon with all sides equal and all interior angles equal).
- Enter Number of Sides (n):
- Input the number of sides your regular polygon has. This must be an integer greater than or equal to 3 (e.g., 3 for a triangle, 4 for a square, 5 for a pentagon, etc.).
- Select Known Parameter:
- Use the “Known Parameter:” dropdown menu to choose which single dimension of the polygon you already know. Options are:
- Side Length (s)
- Inradius / Apothem (r) – the radius of the inscribed circle.
- Circumradius (R) – the radius of the circumscribed circle.
- Area (A)
- Perimeter (P)
- Use the “Known Parameter:” dropdown menu to choose which single dimension of the polygon you already know. Options are:
- Enter Known Value:
- Based on your selection above, an input field will appear (e.g., “Side Length (s)”). Enter the numerical value for this known parameter.
- This value must be positive.
- Enter Units (Optional):
- You can specify the units of your measurements (e.g., cm, m, inches) in the “Units” field. This is for display purposes. Results will be shown with appropriate units (e.g., area in units2).
- Calculate: Click the “Calculate” button.
- Review the Results: The calculator will display:
- Polygon Name: The common name for the polygon if it’s well-known (e.g., Triangle, Square, Pentagon).
- Number of Sides (n): As entered.
- Side Length (s)
- Inradius / Apothem (r)
- Circumradius (R)
- Area (A)
- Perimeter (P)
- Interior Angle (each): The measure of each internal angle of the polygon.
- Exterior Angle (each): The measure of each external angle of the polygon.
- Sum of Interior Angles: The total sum of all internal angles.
- Visual Representation: A diagram of the regular polygon will be shown, scaled appropriately, often with its inradius and circumradius indicated.
- Clear: Click the “Clear” button to reset all input fields and results.
- Error Messages: If your input is invalid (e.g., non-numeric, n
Regular Polygon Calculator: Exploring Symmetry and Shape
What is a Regular Polygon? The Beauty of Equal Sides and Angles
In the vast and fascinating world of geometry, polygons are fundamental building blocks. A polygon is any 2D closed shape made of straight line segments. But when we talk about a regular polygon, we’re referring to a special, highly symmetrical type: a polygon where all its sides are of equal length, AND all its interior angles are of equal measure. Think of an equilateral triangle, a perfect square, a stop sign (an octagon) – these are all examples of regular polygons.
This regularity makes them particularly interesting both aesthetically and mathematically. Their symmetry leads to predictable formulas for their properties, such as area, perimeter, and the measures of their angles and characteristic radii. This calculator is designed to help you explore these properties for any regular n-sided polygon (often called an “n-gon”).
Key Characteristics of Regular Polygons
Every regular polygon is defined by a few key characteristics, which this calculator helps you determine:
- Number of Sides (n): This is the most fundamental property. A polygon must have at least 3 sides.
- Side Length (s): The length of any one of the equal sides.
- Interior Angle: Each angle inside the polygon at a vertex. In a regular polygon, all interior angles are equal. The formula is
(n-2) × 180° / n. - Sum of Interior Angles: The total of all interior angles. For any n-sided polygon (regular or not), this sum is
(n-2) × 180°. - Exterior Angle: The angle formed by one side and the extension of an adjacent side. In a regular polygon, all exterior angles are equal, and each is
360° / n. Note that the interior angle and its corresponding exterior angle at any vertex sum to 180°. - Perimeter (P): The total length of all sides. For a regular polygon,
P = n × s. - Apothem or Inradius (r): This is the radius of the largest circle that can be inscribed within the polygon (an incircle). It’s the perpendicular distance from the center of the polygon to the midpoint of any side.
- Circumradius (R): This is the radius of the circle that passes through all the vertices of the polygon (a circumcircle). It’s the distance from the center of the polygon to any vertex.
- Area (A): The amount of 2D space enclosed by the polygon. There are several formulas, often involving n, s, r, or R. A common one using side length is
A = (n × s²) / (4 × tan(π/n)). Another isA = (P × r) / 2. - Polygon Name: Many regular polygons have common names (e.g., Triangle (n=3), Square (n=4), Pentagon (n=5), Hexagon (n=6), Octagon (n=8), Decagon (n=10)).
Where Do We See Regular Polygons?
Regular polygons are not just abstract geometric concepts; they appear in nature, art, architecture, and technology:
- Nature: Honeycomb cells are hexagonal, snowflakes often exhibit six-fold symmetry, and some crystals form regular polygonal shapes.
- Architecture & Design: Tiling patterns often use regular polygons like squares, triangles, and hexagons. Domes and other structures might incorporate regular polygonal bases.
- Art & Symbolism: Stars are often based on regular polygons (pentagrams, hexagrams). Mandalas and other intricate designs frequently use regular polygonal symmetry.
- Engineering & Manufacturing: Nuts and bolt heads are often hexagonal for easy gripping. Gears can have teeth arranged in a circular, regular pattern.
- Signage: Stop signs are regular octagons, and yield signs are inverted equilateral triangles.
The Math Behind the Calculations: Key Formulas
This calculator uses established geometric formulas to derive the properties of a regular polygon once the number of sides (n) and one other key parameter (like side length ‘s’, inradius ‘r’, or circumradius ‘R’) are known.
For example, if you provide ‘n’ and side length ‘s’:
- Perimeter (P) =
n × s - Inradius (r) =
s / (2 × tan(π/n)) - Circumradius (R) =
s / (2 × sin(π/n)) - Area (A) =
(n × s²) / (4 × tan(π/n))
If you provide a different parameter, say the inradius ‘r’, the calculator first solves for ‘s’ using a rearranged formula (s = 2 × r × tan(π/n)) and then proceeds to calculate all other properties. The trigonometric functions (sine, tangent) are crucial here, operating on angles derived from ‘n’ (specifically π/n radians, which is 180°/n).
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture.” – Bertrand Russell. The symmetry of regular polygons is a perfect example of this mathematical beauty.
Understanding Inradius (Apothem) and Circumradius
These two radii are particularly important for regular polygons:
- Inradius (r – Apothem): Imagine drawing a circle that fits perfectly inside the polygon, touching the midpoint of each side. The radius of this circle is the inradius or apothem. It’s a measure of how “thick” the polygon is from its center to its sides.
- Circumradius (R): Now imagine drawing a circle that passes through every single corner (vertex) of the polygon. The radius of this circle is the circumradius. It’s a measure of how “wide” the polygon is from its center to its corners.
For any regular polygon, the circumradius (R) will always be greater than or equal to the inradius (r). They are equal only in the conceptual limit where n approaches infinity (forming a circle, though a polygon by definition has straight sides).
Visualizing Your Polygon
To make these geometric concepts more tangible, this calculator includes a visual representation of the regular polygon you’re analyzing. Based on the calculated dimensions (usually scaled using the circumradius), it draws the n-sided shape. This can help you see the angles, the relationship between the sides, and how the inradius and circumradius relate to the overall structure. For instance, you’ll see the inradius line extending from the center to the midpoint of a side, and the circumradius from the center to a vertex.
Conclusion: Your Companion for Exploring Regular Polygons
Regular polygons are a fascinating and foundational topic in geometry. Their perfect symmetry makes them both beautiful to look at and rich in mathematical properties. This calculator is designed to be a user-friendly tool for students, educators, designers, engineers, or anyone curious about these shapes. By allowing you to calculate all key properties from minimal input and providing a visual representation, it aims to demystify the geometry of regular polygons and make their exploration accessible and engaging.