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Calculate the area, perimeter, and apothem of a regular octagon from the side length. Uses the formula A = 2(1 + √2) × s².
Area: A = 2(1 + √2) × s² ≈ 4.8284 × s²
Perimeter: P = 8s
Apothem: a = (s/2)(1 + √2) ≈ 1.2071 × s
| Side Length | Area | Perimeter | Apothem | Example |
|---|---|---|---|---|
| 1 cm | 4.83 cm² | 8 cm | 1.21 cm | Small game piece |
| 2 cm | 19.31 cm² | 16 cm | 2.41 cm | Large coin |
| 3 cm | 43.46 cm² | 24 cm | 3.62 cm | Coaster |
| 5 cm | 120.71 cm² | 40 cm | 6.04 cm | Small tile |
| 10 cm | 482.84 cm² | 80 cm | 12.07 cm | Decorative plate |
| 15 cm | 1,086.40 cm² | 120 cm | 18.11 cm | Clock face |
| 20 cm | 1,931.37 cm² | 160 cm | 24.14 cm | Wall mirror |
| 1 in | 4.83 in² | 8 in | 1.21 in | Nut/bolt head |
| 6 in | 173.82 in² | 48 in | 7.24 in | Trivet |
| 12.5 in | 754.44 in² | 100 in | 15.09 in | Standard stop sign |
| 1 ft | 4.83 ft² | 8 ft | 1.21 ft | Small garden bed |
| 2 ft | 19.31 ft² | 16 ft | 2.41 ft | Table top |
| 3 ft | 43.46 ft² | 24 ft | 3.62 ft | Hot tub |
| 5 ft | 120.71 ft² | 40 ft | 6.04 ft | Gazebo |
| 10 ft | 482.84 ft² | 80 ft | 12.07 ft | Large deck |
A regular octagon is an eight-sided polygon (also called an octagon) where all sides have equal length and all interior angles are equal. Each interior angle of a regular octagon measures exactly 135 degrees, and the sum of all interior angles totals 1,080 degrees. The word "octagon" comes from the Greek words "okto" meaning eight and "gonia" meaning angle.
The most universally recognized octagon is the stop sign, which has been the standard octagonal shape for traffic control signs since 1923 in the United States. The octagon was chosen because its distinctive shape is easily recognizable even from behind, helping drivers identify the sign type before they can read the text. A standard stop sign has a side length of 12.5 inches (31.75 cm).
Octagons appear extensively in architecture, art, and design. From the octagonal dome of the Florence Cathedral to octagonal floor tiles, this shape offers a balance between the efficiency of a circle and the practicality of straight edges. Octagonal structures provide excellent structural distribution of forces and maximize interior space while maintaining geometric elegance.
A = 2(1 + √2) × s² ≈ 4.8284 × s²
where s is the side length of the regular octagon
A = 2(1 + √2) × s²
A = 4.8284 × 12.5²
A = 4.8284 × 156.25
A = 754.44 square inches
Perimeter = 8 × 12.5 = 100 inches
A = 4.8284 × 6²
A = 4.8284 × 36
A = 173.82 cm²
You would need about 58 tiles to cover 1 square meter.
A = 4.8284 × 4²
A = 4.8284 × 16
A = 77.25 square feet
This gazebo floor is roughly equivalent to a 9 × 9 foot square room.
For a quick estimate, multiply the side length squared by 5. This gives you a result within about 3.5% of the actual area. For example: side = 10, estimate = 10² × 5 = 500 (actual: 482.84).
| Side | Area | Perimeter | Apothem | Long Diagonal |
|---|---|---|---|---|
| 1 | 4.83 | 8 | 1.21 | 2.61 |
| 2 | 19.31 | 16 | 2.41 | 5.23 |
| 3 | 43.46 | 24 | 3.62 | 7.84 |
| 4 | 77.25 | 32 | 4.83 | 10.45 |
| 5 | 120.71 | 40 | 6.04 | 13.07 |
| 10 | 482.84 | 80 | 12.07 | 26.13 |
| 15 | 1,086.40 | 120 | 18.11 | 39.20 |
| 20 | 1,931.37 | 160 | 24.14 | 52.26 |
| Shape | Sides | Area (s=10) | Interior Angle |
|---|---|---|---|
| Square | 4 | 100.00 | 90° |
| Pentagon | 5 | 172.05 | 108° |
| Hexagon | 6 | 259.81 | 120° |
| Heptagon | 7 | 363.39 | 128.57° |
| Octagon | 8 | 482.84 | 135° |
| Decagon | 10 | 769.42 | 144° |
| Dodecagon | 12 | 1,115.26 | 150° |
Stop signs worldwide use the octagonal shape. Understanding octagon dimensions helps in sign manufacturing, placement planning, and visibility calculations.
Octagonal designs appear in gazebos, towers, fountains, and window frames. Architects need precise area calculations for materials and structural planning.
Octagonal tiles combined with small square tiles create classic floor patterns. Area calculations determine how many tiles are needed for a given space.
Octagonal cross-sections appear in bolts, nuts, and structural columns. Precise geometry is essential for manufacturing tolerances and load calculations.
The area of a regular octagon is approximately 4.8284 times the side length squared. Memorize this constant for quick calculations.
You can also calculate area as A = (1/2) × perimeter × apothem. This serves as a good cross-check for your primary calculation.
The formula A = 2(1 + √2)s² only works for regular octagons where all sides and angles are equal. Irregular octagons require different methods.
The long diagonal of an octagon is about 2.613 times the side length. Using the diagonal as the side length will drastically overestimate the area.
Each regular polygon has its own area multiplier. The hexagon uses ≈ 2.598, the octagon uses ≈ 4.828. Mixing these up gives incorrect results.
If the side length is in inches, the area will be in square inches. Converting units after calculation requires squaring the conversion factor (e.g., 1 ft² = 144 in²).
A regular octagon is an eight-sided polygon where all sides are equal in length and all interior angles are equal at 135 degrees each. It is one of the most recognized polygons due to its use in stop signs worldwide.
The area of a regular octagon with side length s is A = 2(1 + the square root of 2) times s squared, which simplifies to approximately 4.8284 times s squared. This formula calculates the entire interior area of the octagon.
Each interior angle of a regular octagon measures exactly 135 degrees. The sum of all interior angles is 1,080 degrees, calculated by the formula (n-2) times 180 where n equals 8.
The perimeter of a regular octagon is simply 8 times the side length (P = 8s). Since all eight sides are equal, you multiply the length of one side by 8.
The apothem is the perpendicular distance from the center of the octagon to the midpoint of any side. For a regular octagon with side length s, the apothem equals s times (1 + the square root of 2) divided by 2, or approximately 1.2071 times s.
A standard stop sign in the United States has a side length of 12.5 inches (31.75 cm), resulting in an octagonal area of approximately 754.4 square inches or 4,868 square centimeters. The total width from flat to flat is about 30 inches.
Regular octagons alone cannot tile a plane because they leave gaps. However, combining regular octagons with small squares creates a popular tiling pattern called the truncated square tiling, commonly seen in bathroom and kitchen floors.
A regular octagon has all eight sides equal and all eight angles equal at 135 degrees. An irregular octagon has eight sides but they can be different lengths and angles can vary, as long as the total of interior angles equals 1,080 degrees.
For a regular octagon with longest diagonal d (corner to corner), the side length s equals d divided by (1 + the square root of 2). Once you know the side length, use the standard area formula A = 2(1 + the square root of 2) times s squared.
Octagons appear frequently in architecture including baptisteries, church domes, towers, gazebos, and decorative windows. The octagonal shape provides structural strength while maximizing interior space, and has been used since ancient Roman and Byzantine architecture.
This calculator is provided for educational and informational purposes only. Always verify critical calculations independently.