Calculate properties of regular polygons
Last updated: March 2026
Perimeter:
P = 6 × 5 = 30 units
Area:
A = 64.951905 square units
Interior Angle:
120°
Exterior Angle:
60°
A regular polygon is a geometric shape with equal-length sides and equal interior angles. Examples include equilateral triangles (3 sides), squares (4 sides), regular pentagons (5 sides), and regular hexagons (6 sides). The key property is that all vertices lie on a circle, making regular polygons highly symmetrical and useful in design, architecture, and mathematics.
Regular polygons are fundamental in geometry because they possess predictable mathematical relationships. The interior angles, perimeter, area, and radius all have precise formulas based on the number of sides. Understanding these relationships is essential for architecture, engineering, and advanced mathematics, where regular polygons appear frequently in tessellations and circular systems.
Enter the number of sides
Specify how many sides your regular polygon has. The minimum is 3 (triangle); there is no practical maximum. The calculator works for all regular polygons.
Why: The number of sides fundamentally determines all angles and properties of a regular polygon. From n, we can calculate interior angles and derive all other properties.
Enter the side length
Input the length of one side in your preferred units (cm, inches, meters, etc.). In a regular polygon, all sides are equal, so you only need to specify one length.
Why: Side length scales all calculated dimensions proportionally. Combined with the number of sides, it determines perimeter, area, and radius measurements.
Review Properties tab results
The Properties tab displays perimeter, area, interior angle, exterior angle, and basic measurements. These are the most commonly needed values for design and construction.
Why: Perimeter tells you total edge length, area tells you coverage, and angles are essential for understanding how edges connect at vertices.
Explore Advanced tab for geometric relationships
Switch to the Advanced tab to access circumradius (radius of outer circle), apothem (inradius—radius of inner circle), and diagonal count. These reveal deeper geometric structure.
Why: Circumradius and apothem describe the polygon's inscribed and circumscribed circles, essential for 3D structures, and diagonal count is fundamental to understanding polygon connectivity.
Apply results to your project
Use perimeter for material estimation, area for coverage calculations, angles for cutting and fitting, and radii for circular mounting or tessellation planning. Different properties serve different purposes.
Why: Understanding which property solves which problem makes you efficient and prevents costly construction or design errors.
An interior designer is planning a modern bathroom renovation with a distinctive hexagonal tile pattern. Each tile is a regular hexagon with 10 cm sides. The designer needs to determine how much floor space each individual tile covers, calculate the total number of tiles needed for a 2 m × 3 m bathroom floor, understand what cutting angles will be required at edges and corners, and estimate material costs based on area coverage.
Step 1 — Define polygon parameters:
Regular hexagon with n = 6 sides and side length s = 10 cm.
Step 2 — Calculate perimeter:
Perimeter = 6 × 10 cm = 60 cm (total edge length of one tile)
Step 3 — Calculate area per tile:
Area ≈ 259.81 cm² (amount of bathroom floor covered by each tile)
Step 4 — Determine cutting angles:
Interior angle = 120°, Exterior angle = 60° (angles at each vertex for fitting and cutting)
Step 5 — Calculate tiles needed and estimate materials:
Bathroom floor area = 2 m × 3 m = 60,000 cm². Tiles needed ≈ 60,000 ÷ 259.81 ≈ 231 complete tiles (plus extras for cuts).
Check: Interior angle = (6-2) × 180° ÷ 6 = 720° ÷ 6 = 120° ✓. Verify area by alternative calculation: A = (3√3/2) × s² = (3√3/2) × 100 ≈ 259.81 cm² ✓.
Per hexagonal tile: Area = 259.81 cm², Interior angle = 120°
Tiles for bathroom: approximately 231 complete tiles. Each tile touches 6 neighbors at 120° angles, creating a tight, tessellating pattern with no gaps.
The 120° interior angle tells the designer that when two hexagonal tiles meet, they fit together at exactly 120°, which is why hexagons tessellate perfectly (three hexagons meet at each vertex: 3 × 120° = 360°). The designer can order tiles with confidence, knowing the exact cutting angles needed for the bathroom perimeter. Material suppliers can estimate cost: if tiles cost $5 per unit at 259.81 cm² each, the designer budgets for ~231 tiles plus 5-10% waste allowance. The hexagonal pattern creates an aesthetically pleasing, mathematically efficient floor that maximizes coverage with minimal cutting required.
Interior angles are inside the polygon; they sum to (n−2)×180°. Exterior angles are outside at each vertex; they always sum to 360°.
The apothem is the perpendicular distance from the center to the midpoint of any side. It's used to calculate area: A = (Perimeter × Apothem) / 2.
The circumradius is the distance from the center to any vertex. All vertices of a regular polygon lie exactly on a circle with this radius.
No. In a regular polygon, diagonals connecting vertices at the same distance from the center have equal length, but different diagonals can vary.
Interior angle = (n−2) × 180° / n. As n increases, interior angles approach 180°, approaching a circle.
A square (4 sides). Formula check: (4−2) × 180° / 4 = 90°. Hexagons have 120°, pentagons have 108°.
No, this calculator is for regular polygons with equal sides and angles. Irregular polygons require different methods.
They appear in nature (honeycombs, snowflakes), architecture (domes, tiles), and mathematics (group theory, tessellations). They're fundamental to symmetry.
Specific calculations for 5-sided polygons
Specific calculations for 8-sided polygons
Infinite-sided polygon properties
Irregular quadrilateral properties
3-sided polygon properties
Calculate perimeter for any shape
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