Polygon Square Footage Calculator — Regular Multi-Sided Shapes
Regular polygons — shapes with equal sides and equal angles — appear in octagonal pools, hexagonal gazebo decks, pentagonal patio designs, and decorative tile medallion centres. The area of a regular polygon requires knowing the number of sides and either the side length or the apothem (the perpendicular distance from the centre to the midpoint of any side). This calculator handles both inputs so you are not forced to derive the apothem when you only have the side length.
The formula for a regular n-sided polygon with side length s is: Area = (n × s² × cot(π/n)) / 4. Our calculator performs this automatically. Enter side count and side length — or perimeter and apothem — for instant results.
Regular Polygon Reference Table
- Equilateral triangle (3 sides): Area = (√3/4) × s² — 1 ft side = 0.433 sq ft
- Square (4 sides): Area = s² — most common, used for all room calculations
- Pentagon (5 sides): Area = 1.720 × s² — 10 ft side = 172 sq ft
- Hexagon (6 sides): Area = 2.598 × s² — 10 ft side = 259.8 sq ft
- Heptagon (7 sides): Area = 3.634 × s² — 10 ft side = 363.4 sq ft
- Octagon (8 sides): Area = 4.828 × s² — 10 ft side = 482.8 sq ft
- Decagon (10 sides): Area = 7.694 × s² — 10 ft side = 769.4 sq ft
- Dodecagon (12 sides): Area = 11.196 × s² — 10 ft side = 1,119.6 sq ft
Octagonal Rooms and Features
Octagonal gazebos are the most common regular polygon structure in residential landscaping. Standard prefabricated gazebo kits come in sizes from 8 ft to 16 ft measured as the octagon's side-to-side diameter (not side length). For an 8-sided gazebo with 6 ft side lengths: Area = 4.828 × 36 = 173.8 sq ft. A 10-side-length octagonal gazebo covers 482.8 sq ft — roughly 2.5 times more than it appears.
Hexagonal Tile Patterns
Hexagonal tiles are measured differently from square or rectangular tiles because they interlock without grout lines on shared edges. A field of hexagonal tiles with a flat-to-flat dimension of 4 inches has an area per tile of approximately 13.86 square inches, meaning you need about 10.4 tiles per square foot. For the floor area, calculate total square footage normally, then divide by the per-tile area to get the tile count. Add 15% for waste because hexagonal offcuts at walls are irregular and non-reusable.
Polygon Decks and Patios
Octagonal and hexagonal decks require more complex framing than rectangular decks, but the area calculation for materials (decking boards, concrete footings) uses the polygon formula. Composite decking boards ordered for a 173 sq ft octagonal deck need 10% waste added (190 sq ft total) because of the 45-degree angled cuts at the perimeter.
Related tools: circle calculator · trapezoid calculator · irregular shape · triangle calculator
Regular Polygon Pergolas, Gazebos, and Decks
Hexagonal and octagonal outdoor structures are common in residential landscaping. A hexagonal pergola with 8 ft side lengths has an area of 2.598 × 64 = 166.3 sq ft. An octagonal deck with 6 ft side lengths: 4.828 × 36 = 173.8 sq ft. For composite decking boards on an octagonal deck, order 173.8 sq ft plus 15% waste (199.9 sq ft ≈ 200 sq ft). Deck frame lumber is ordered by linear foot — calculate the perimeter (8 × 6 ft = 48 linear ft for the outer rim joist) and the joist spacing to determine the number of 2×8 or 2×10 joists needed. Concrete footings for a hexagonal gazebo require one footing per corner (6 footings), each typically 12 inches in diameter and 36 inches deep, holding approximately 0.25 cubic yards of concrete each.
Polygon Area in Tile Medallion Design
Custom tile medallions — decorative centrepieces set into a tile floor — are frequently hexagonal, octagonal, or multi-sided geometric forms. Calculating the medallion area tells you how many pieces of specialty tile the fabricator needs to cut. A 24-inch octagonal medallion (side length approximately 9.9 inches) has an area of 4.828 × (0.825 ft)² = 3.28 sq ft. Surrounding this medallion with standard 12×12 field tile requires subtracting the 3.28 sq ft medallion footprint from the room area before ordering field tile. Order the medallion and field tile separately — they come from different sources and are installed in a specific sequence (field tile first, medallion last, centred on the layout).