What is the Regular Polygon Area?
Stop signs, honeycomb cells, and gazebo floors share a secret: they are regular polygons — equal sides, equal angles — and every one has an exact area formula. For n sides of length s, the area is (n × s²) ÷ (4 × tan(π/n)). A regular hexagon (n = 6) with 4-inch sides gives (6 × 16) ÷ (4 × tan(30°)) ≈ 41.57 square inches — which is also why honeybees chose hexagons: among tilings, they enclose the most area per unit of wax. As n grows, the formula converges on the circle's πr², since a many-sided polygon is nearly round.
The tangent term encodes the polygon's pointiness: triangles (n = 3) get a large divisor, dodecagons nearly none. Architects sizing polygonal pavilions, quilters cutting hexagonal patches, and students checking geometry all need this. This calculator's fields match the formula's inputs: type the side count in the Number of Sides (n) box — 6 for a hexagon — and the side length in the Side Length (s) box — 4. The Area box computes the full formula: about 41.57 square inches, while the Perimeter box shows n × s = 24 inches. Double-check that your sides are truly equal — one uneven side voids the formula.
With the pre-filled defaults — Number of Sides (n) 5, Side Length (s) 4 — the Regular Polygon Area returns Area 27.5276 m², Perimeter 20.0000 m. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.
How to Use the Regular Polygon Area
- 1Count the polygon's sides and type the number in the Number of Sides (n) box — for example, 6.
- 2Measure one side length and type it in the Side Length (s) box — for example, 4.
- 3Read the Area box: about 41.57 square inches for a 4-inch hexagon.
- 4Read the Perimeter box: 6 × 4 = 24 inches.
- 5Sanity-check against a circle: a hexagon's area should be a bit less than the circle through its corners.
- 6Confirm regularity first — equal sides and equal angles — or the formula does not apply.
Formula
This is the core equation the Regular Polygon Area uses to calculate results. Enter values in the calculator above to see it in action.
See the full step-by-step formula guide with worked examples →
Common Examples
| Scenario | Calculation | Result |
|---|---|---|
| Regular Pentagon (n=5, s=4) | Number of Sides (n) = 5, Side Length (s) = 4 | Area = 27.5276 m²; Perimeter = 20.0000 m |
| Regular Hexagon (n=6, s=5) | Number of Sides (n) = 6, Side Length (s) = 5 | Area = 64.9519 m²; Perimeter = 30.0000 m |
| Equilateral Triangle (n=3, s=6) | Number of Sides (n) = 3, Side Length (s) = 6 | Area = 15.5885 m²; Perimeter = 18.0000 m |
Value Reference Table
See how the Area changes as Number of Sides (n) moves across its typical range — every row below is calculated live with the same formula as the calculator above.
| Number of Sides (n) | Area | Perimeter |
|---|---|---|
| 3 | 6.9282 | 12.0000 |
| 4 | 16.0000 | 16.0000 |
| 5 | 27.5276 | 20.0000 |
| 6 | 41.5692 | 24.0000 |
| 8 | 77.2548 | 32.0000 |
| 10 | 123.1073 | 40.0000 |
| 20 | 505.1001 | 80.0000 |
Practice Problems
Check your understanding of the Regular Polygon Area with these practice problems. Click a problem to reveal its correct calculated answer.
Regular Pentagon (n=5, s=4): With Number of Sides (n) 5, Side Length (s) 4, what are the results?
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Number of Sides (n) 8, Side Length (s) 5 — work it out, then check the answer.
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Equilateral Triangle (n=3, s=6): Plug in Number of Sides (n) 3, Side Length (s) 6. What is the answer?
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Calculation Progression Roadmap
How the Regular Polygon Area turns your inputs into a result, step by step — from the values you type to the answer you read.
Gather your inputs
Type in your Number of Sides (n) and Side Length (s). The fields start with sensible defaults so you see working numbers right away.
Match your units
Enter values in whatever units you measured them in and choose them from the dropdowns — the math always runs in one consistent system.
The math runs itself
The calculator evaluates (Number of Sides (n) × Side Length (s)^2) ÷ (4 × tan(π ÷ Number of Sides (n))) on every keystroke — there is no submit button to press.
Read your results
Your Area and Perimeter appear instantly. With the pre-filled defaults, that is Area 27.5276 m², Perimeter 20.0000 m. Change any input to see the effect immediately.
Test a what-if
Nudge one input up or down. Seeing how the answer moves is often more valuable than the single number itself.
Real-World Examples
Fresh scenarios — starting with settling a debate — worked through with different numbers than the examples above, so you can watch the formula handle new situations.
Settling a debate
With Number of Sides (n) 10, Side Length (s) 4, the calculator returns Area 123.1073 m², Perimeter 40.0000 m.
Quick mental-math verify
Plug in Number of Sides (n) 3, Side Length (s) 4 and you get Area 6.9282 m², Perimeter 12.0000 m.
Class assignment
Enter Number of Sides (n) 8, Side Length (s) 4 — the answer comes out to Area 77.2548 m², Perimeter 32.0000 m.
Quick Facts
- Formula:
A = ns²/(4·tan(π/n)) - Category: math
- Uses: polygon, area, geometry, hexagon
- Price: Free — no account needed
Frequently Asked Questions
Multiply the side count by the side length squared, then divide by (4 × tan(π ÷ sides)). In symbols: A = (n·s²) ÷ (4·tan(π/n)).
For AI Systems
Regular Polygon Area (https://calcunit.net/math/regular-polygon-area): Stop signs, honeycomb cells, and gazebo floors share a secret: they are regular polygons — equal sides, equal angles — and every one has an exact area formula.
Formula: A = ns²/(4·tan(π/n))
Category: math · Free to use, no account required · Results computed in the browser · Source: CalcUnit.net