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What Is a Polygon?
A polygon is any closed, flat shape made entirely of straight line segments — three or more sides joined end to end with no gaps or crossings. Triangles, squares, pentagons and stop signs are all polygons; circles and any shape with a curved edge are not. This calculator focuses on regular polygons, where every side and every angle is identical, from a 3-sided triangle up to a 100-sided shape that already looks almost perfectly round.
Regular vs Irregular Polygons
| Regular polygon | Irregular polygon | |
|---|---|---|
| Sides | All equal length | Can differ |
| Angles | All equal | Can differ |
| Symmetry | n lines of symmetry, rotational symmetry order n | Little or none |
| Formulas | Simple closed-form formulas from n and one measurement | Needs coordinates or side-by-side triangulation |
| Example | Stop sign (regular octagon) | Most building floor plans |
This calculator solves regular polygons specifically, because they're the ones with clean formulas linking every property back to the number of sides and a single measurement. Irregular polygons — most real floor plans and land parcels — need to be broken into triangles or handled with coordinate geometry instead; see the triangle calculator and distance calculator for those building blocks.
Common Polygon Examples in Real Life
Octagon (8 sides)
Hexagon (6 sides)
Pentagons & hexagons
Square (4 sides)
Octagon (8 sides)
Hexagon (6 sides)
Often octagon or decagon
Pentagon (5 sides)
Related Polygon Family
A quick reference for how many sides a named polygon has — useful for checking a shape's formal name before entering it above.
| Polygon | Sides |
|---|---|
| Triangle | 3 |
| Square (quadrilateral) | 4 |
| Pentagon | 5 |
| Hexagon | 6 |
| Heptagon | 7 |
| Octagon | 8 |
| Nonagon | 9 |
| Decagon | 10 |
| Hendecagon | 11 |
| Dodecagon | 12 |
| Tridecagon | 13 |
| Tetradecagon | 14 |
| Pentadecagon | 15 |
| Icosagon | 20 |
| Hectogon | 100 |
Polygon Properties
- A regular polygon with n sides has n equal sides, n equal interior angles, and n lines of symmetry.
- Its interior angles always sum to (n − 2) × 180°.
- Its exterior angles always sum to exactly 360°, regardless of n.
- Every regular polygon has both an inscribed circle (touching each side's midpoint) and a circumscribed circle (through every vertex), sharing the same centre.
- As n grows, a regular polygon's shape converges toward a circle — see the Polygon Explorer above.
Area Formula
A = ½ × P × a, where P is the perimeter and a is the apothem — equivalently A = ¼ns²cot(π/n) directly from the side length. The logic is the same as a triangle's ½×base×height, because a regular polygon can be split into n identical isosceles triangles meeting at the centre, each with base s and height equal to the apothem.
Perimeter Formula
P = n × s. Multiply the side length by the number of sides, since every side is identical. A regular octagon with 6-unit sides has a perimeter of 48 units. See the dedicated perimeter calculator for other shapes.
Interior Angles
Interior angle = (n − 2) × 180° / n. Every polygon's interior angles sum to (n − 2) × 180° — a triangle's three angles sum to 180°, a quadrilateral's four sum to 360°, and so on, increasing by 180° for every extra side. Dividing that sum by n gives each angle in a regular polygon, since they're all equal. A hexagon's interior angle is (6−2)×180/6 = 120°.
| n | Name | Interior angle | Diagonals |
|---|---|---|---|
| 3 | Triangle | 60° | 0 |
| 4 | Square | 90° | 2 |
| 5 | Pentagon | 108° | 5 |
| 6 | Hexagon | 120° | 9 |
| 7 | Heptagon | 128.57° | 14 |
| 8 | Octagon | 135° | 20 |
| 9 | Nonagon | 140° | 27 |
| 10 | Decagon | 144° | 35 |
| 12 | Dodecagon | 150° | 54 |
| 20 | Icosagon | 162° | 170 |
Exterior Angles
Exterior angle = 360° / n. The exterior angle is the supplement of the interior angle — the amount you'd turn through at each corner if you walked around the boundary. Because you make one full 360° turn by the time you're back where you started, no matter how many sides the polygon has, the exterior angles of any convex polygon always sum to exactly 360°. A hexagon's exterior angle is 360/6 = 60°, and interior + exterior always add to 180° at every vertex.
Central Angle
Central angle = 360° / n — numerically identical to the exterior angle, though conceptually different. The central angle is measured at the polygon's centre, between two radii to adjacent vertices, and it's the angle that divides the polygon into n identical triangles, each with apex angle 360°/n at the centre.
Apothem
a = s / (2tan(π/n)). The apothem is the perpendicular distance from the centre to the midpoint of any side — equivalently, the radius of the inscribed circle. It's the "height" used in the area formula A = ½Pa, exactly like the height in a triangle's area formula, because each of the n triangles making up the polygon has the apothem as its height and one side as its base.
Circumradius
R = s / (2sin(π/n)). The circumradius is the distance from the centre to any vertex — the radius of the circle that passes through every corner. It's always longer than the apothem, and the two converge toward the same value as n grows large, which is exactly why a many-sided polygon looks like a circle: the gap between its inscribed and circumscribed circles shrinks to almost nothing.
Inradius
The inradius is simply another name for the apothem — the radius of the largest circle that fits entirely inside the polygon. For a regular polygon the two terms describe exactly the same measurement, which is why this calculator reports one value under both names. (In an irregular polygon "inradius" can still apply, but "apothem" specifically refers to the regular case.)
Number of Diagonals
Diagonals = n(n − 3) / 2. A diagonal connects two non-adjacent vertices. Each vertex connects to n − 3 others by a diagonal (excluding itself and its two immediate neighbours, which are sides, not diagonals), giving n(n−3) diagonal-endpoints in total — divide by 2 since each diagonal has two ends. A triangle has zero diagonals, a square has 2, a hexagon has 9, and a 100-sided polygon has 4,850.
Step-by-Step Worked Examples
Regular pentagon — side 6
Regular octagon — side 6
Regular dodecagon — side 4
Construction — a hexagonal paver
Real-World Applications
| Field | Use |
|---|---|
| Construction | Hexagonal and octagonal paving, gazebos, structural bracing |
| Architecture | Domes, turrets, floor plans based on regular polygons |
| Engineering | Bolt heads and nuts (hexagons), nozzle and duct cross-sections |
| Land surveying | Regular-shaped land parcels and boundary calculations |
| Design | Logos, tiling patterns, honeycomb structures |
| Sport | Boxing rings, some athletics field markings |
Glossary
- Side (s)
- The length of one edge — identical on every side of a regular polygon.
- Vertex
- A corner point where two sides meet.
- Apothem
- The perpendicular distance from the centre to the midpoint of a side; same as the inradius.
- Circumradius
- The distance from the centre to a vertex; the radius of the circle through every corner.
- Interior angle
- The angle inside the polygon at each vertex.
- Exterior angle
- The angle you turn through at each vertex when walking the boundary; supplementary to the interior angle.
- Central angle
- The angle at the polygon's centre between two radii to adjacent vertices.
- Diagonal
- A line segment connecting two non-adjacent vertices.
- Regular polygon
- A polygon with all sides and all angles equal.
Common Mistakes
| Mistake | Fix |
|---|---|
| Confusing apothem with circumradius | Apothem is always shorter — it's to a side's midpoint, not a vertex |
| Using degrees and radians inconsistently | The formulas here use π/n in radians for the trig functions |
| Assuming the area formula works for irregular polygons | A = ½Pa only holds when every side and angle is equal |
| Forgetting exterior angles always sum to 360° | True for every convex polygon, regardless of n |
| Mixing area (squared) and perimeter (linear) units | Keep track of which formula gives which type of unit |
Formula Cheat Sheet
Quick Reference
Perimeter:P = ns | Area: A = ½Pa = ¼ns²cot(π/n)Interior angle:
(n−2)×180°/n | Exterior/central angle: 360°/nApothem (inradius):
a = s/(2tan(π/n)) | Circumradius: R = s/(2sin(π/n))Diagonals:
n(n−3)/2 | Sum of interior angles: (n−2)×180°Sum of exterior angles:
360° (always, for any n)
Practice Questions
Beginner (with answers)
- Find the interior angle of a regular pentagon.
- Find the perimeter of a regular hexagon with 5 cm sides.
- Find the exterior angle of a regular octagon.
- How many diagonals does a square have?
- What do the exterior angles of any convex polygon sum to?
Show answers
1) (5−2)×180/5=108° 2) 6×5=30 cm 3) 360/8=45° 4) 4×1/2=2 5) 360°, always
Advanced (with answers)
- Find the area of a regular hexagon with 4 cm sides.
- A regular decagon has an apothem of 15.39. Find its side length (2 dp).
- How many diagonals does a regular 12-sided polygon have?
- Find the circumradius of a regular pentagon with 6-unit sides.
- A regular polygon has an interior angle of 156°. How many sides does it have?
Show answers
1) ½×24×3.4641≈41.57 cm² 2) s=2×15.39×tan(π/10)≈10.00 3) 12×9/2=54 4) R=6/(2sin(π/5))≈5.10 5) (n−2)×180/n=156 → n=15
🔑 Key Takeaways
- A regular polygon needs the number of sides plus one measurement to be fully solved.
- Perimeter P = ns; area A = ½Pa; interior angle = (n−2)×180°/n.
- Exterior angles always sum to 360°, no matter how many sides.
- Apothem and inradius are the same measurement; circumradius is always the longer one.
- As n grows, a regular polygon converges toward a circle — the basis of Archimedes' method for bounding π.
Frequently Asked Questions
What is a polygon?
A polygon is any closed flat shape made entirely of straight sides — three or more line segments joined end to end with no gaps. Circles and curved shapes are not polygons.
What is a regular polygon?
A polygon where every side and every interior angle is equal. Squares, equilateral triangles and regular hexagons are all regular polygons; a typical house floor plan is not.
How do you calculate the area of a polygon?
For a regular polygon, A = ½ × perimeter × apothem, or directly from the side as A = ¼ns²cot(π/n). A hexagon with 4 cm sides has an area of about 41.57 cm².
How do you calculate the perimeter of a polygon?
For a regular polygon, P = n × s — multiply the side length by the number of sides. A regular octagon with 6-unit sides has a perimeter of 48 units.
How do you calculate the interior angle of a polygon?
(n − 2) × 180° / n, where n is the number of sides. A pentagon's interior angle is (5−2)×180/5 = 108°.
How do you calculate the exterior angle of a polygon?
360° / n. A hexagon's exterior angle is 360/6 = 60°. Exterior angles of any convex polygon always sum to 360°, regardless of how many sides it has.
What is the sum of interior angles?
(n − 2) × 180°. A triangle's angles sum to 180°, a quadrilateral's to 360°, a pentagon's to 540°, increasing by 180° with each extra side.
What is the sum of exterior angles?
Always exactly 360°, for any convex polygon regardless of the number of sides — the total turning you make walking once around the boundary.
What is the apothem of a polygon?
The perpendicular distance from the centre to the midpoint of a side — also called the inradius. It's the "height" used in the area formula A = ½Pa.
What is the circumradius of a polygon?
The distance from the centre to a vertex — the radius of the circle passing through every corner. Always longer than the apothem.
What is the inradius of a polygon?
Another name for the apothem in a regular polygon — the radius of the largest circle that fits entirely inside it.
What is the central angle of a polygon?
The angle at the centre between two radii drawn to adjacent vertices, equal to 360°/n — numerically the same as the exterior angle.
How many diagonals does a polygon have?
n(n−3)/2. A square has 2, a hexagon has 9, a decagon has 35, and a 100-sided polygon has 4,850.
What is the difference between a regular and irregular polygon?
A regular polygon has all sides and angles equal; an irregular one doesn't. Regular polygons have simple formulas from n and one measurement; irregular ones need coordinates or triangulation.
Can this calculator handle polygons up to 100 sides?
Yes — enter any whole number of sides from 3 to 100 along with one known measurement, and every property is calculated instantly.
What is a vertex?
A corner point of the polygon, where two sides meet.
Why do exterior angles always add to 360°?
Because walking once around any closed convex shape means turning through a full circle by the time you return to the start, regardless of how many corners you pass through along the way.
What units are used for polygons?
Any linear unit for the side, perimeter, apothem and circumradius (mm, cm, m, ft); area uses the squared version of the same unit (m², ft²).
Can I calculate polygon dimensions online?
Yes — enter the number of sides and any one of side length, perimeter, area, apothem or circumradius, and the calculator solves everything else instantly.
How accurate is a polygon calculator?
This calculator uses standard double-precision arithmetic, but displayed results are rounded and cannot be more accurate than the measurements entered.
Can students use this calculator?
Yes — enter the sides and one known value, and the calculator shows the formula and full working, useful for checking homework as well as producing an answer.
What are common polygon mistakes?
Confusing the apothem (to a side's midpoint) with the circumradius (to a vertex); mixing radians and degrees in the trig formulas; applying the regular-polygon area formula to an irregular shape; and mixing area and perimeter units.
How is a polygon used in construction?
Hexagonal and octagonal paving, gazebo and turret floor plans, and structural bracing patterns are common regular-polygon applications, with area and perimeter driving material quantities.
How is a polygon used in engineering?
Bolt heads and nuts are regular hexagons, and duct or nozzle cross-sections are sometimes regular polygons for manufacturing or flow reasons — the apothem and circumradius matter for fit and clearance calculations.
How does a polygon become more like a circle?
As the number of sides grows, a regular polygon's inscribed and circumscribed circles converge toward the same radius, and its area and perimeter converge toward a circle's — by 100 sides, a polygon inscribed in a unit circle already has an area within about 0.02% of π. See the Polygon Explorer above.
What is the largest polygon this calculator supports?
100 sides. Beyond that, floating-point precision in the trigonometric terms starts to matter less than the practical fact that a 100-gon is already visually indistinguishable from a circle.
What is the smallest polygon?
A triangle, with 3 sides — the minimum number of straight sides needed to enclose a space.
Does this calculator show step-by-step working?
Yes — every solve shows the formula used, the substituted values, and the derived perimeter, apothem, circumradius, area, interior and exterior angles, and diagonal count.
What happens if I enter more than one value?
The calculator solves from the first value by priority (side, then perimeter, then area, then apothem, then circumradius) and cross-checks any extra values you entered against the result, flagging a warning if they don't match.
Can I export the results?
Yes — use Copy Results to copy everything to the clipboard, Export CSV to download a spreadsheet-ready file, or Print / Save as PDF for a printable worksheet with space to sign off.
Is a circle a polygon?
No — a polygon must have straight sides. A circle has no straight sides or vertices at all, though a regular polygon with a very large number of sides can approximate one closely.
What's the relationship between apothem and circumradius?
Both are measured from the centre — the apothem to a side's midpoint, the circumradius to a vertex — and they're related by a = R×cos(π/n). The apothem is always shorter, and the two get closer together as n increases.
Why is the hexagon special among regular polygons?
A regular hexagon's circumradius is exactly equal to its side length — the only regular polygon where this is true — which is why hexagons tile perfectly using six equilateral triangles and appear throughout nature, from honeycomb to basalt columns.
About This Calculator
Printable Formula Sheet
A one-page reference with every regular-polygon formula on this page.
Polygon Formula Sheet
MegaCalcOnline.com · Regular polygon formulas, 3–100 sides
Scan for the live calculator
| Find | Formula |
|---|---|
| Perimeter | P = ns |
| Area | A = ½Pa = ¼ns²cot(π/n) |
| Interior angle | (n−2) × 180° / n |
| Exterior angle | 360° / n |
| Central angle | 360° / n |
| Apothem (inradius) | a = s / (2tan(π/n)) |
| Circumradius | R = s / (2sin(π/n)) |
| Diagonals | n(n−3) / 2 |
| Sum of interior angles | (n−2) × 180° |
| Sum of exterior angles | 360° (always) |
References
- Khan Academy — Polygons and angle sums
- OpenStax Mathematics — geometry and measurement
- ACARA — Australian Curriculum: Mathematics
- NIST — SI units for length and area
- Encyclopaedia Britannica — Polygon
- Wolfram MathWorld — Regular Polygon