units
Please enter a positive value.
Enter your measurement in the units box below
What measurement are you entering?
Area of Regular Octagon
—
Was this calculator helpful?
Was this calculator helpful?
Sources & Methodology
Formulas verified against standard geometry references from Khan Academy and Math Is Fun, confirmed with CRC Handbook of Mathematical Sciences octagon constants.
Khan Academy — Regular Polygons
Interior angle formula (n−2)×180°/n for regular polygons; area derivation by triangulation from center
Math Is Fun — Octagon
Area = 2(1+√2)s², apothem, circumradius, and width formulas with numerical constants verified
Methodology: Side input: s is used directly. Circumradius input: s = R × 2sin(π/8). Inradius/apothem input: s = a × 2tan(π/8). Area = 2(1+√2) × s² ≈ 4.8284 × s². Perimeter = 8s. Apothem = s(1+√2)/2. Circumradius = s / (2sin(π/8)). Width across flats = s(1+√2). Width across corners = s√(4+2√2). All results rounded to 4 decimal places.
⏱ Last reviewed: March 2026
How to Calculate the Area of a Regular Octagon
A regular octagon is an 8-sided polygon where all sides are equal in length and all interior angles are equal at 135°. The area formula comes from dividing the octagon into triangles and rectangles from the center, and combining their areas into one elegant constant multiplied by the side length squared.
Area Formula — From Side Length
Area = 2(1 + √2) × s² ≈ 4.8284 × s²
Where s is the side length of any one side.
Example: s = 5 units
Area = 2(1+1.4142) × 25 = 2 × 2.4142 × 25 = 120.71 sq units
Example: s = 5 units
Area = 2(1+1.4142) × 25 = 2 × 2.4142 × 25 = 120.71 sq units
Area Formula — From Circumradius
s = R × 2sin(π/8) ≈ R × 0.7654
Then: Area = 2(1+√2) × s²
Where R is the circumradius (center to vertex). For R = 7:
s = 7 × 0.7654 = 5.358 → Area = 4.8284 × 28.71 = 138.58 sq units
s = 7 × 0.7654 = 5.358 → Area = 4.8284 × 28.71 = 138.58 sq units
Other Key Measurements
Perimeter = 8 × s
Apothem = s × (1+√2) / 2 ≈ s × 1.2071
Width across flats = s × (1+√2) ≈ s × 2.4142
Width across corners = s × √(4+2√2) ≈ s × 2.6131
The apothem is the perpendicular distance from center to any side. Width across flats is the distance between two opposite parallel sides. Width across corners is the longest diagonal.
Regular Octagon Reference Table
| Side (s) | Area | Perimeter | Apothem | Width (flats) |
|---|---|---|---|---|
| 1 | 4.828 | 8 | 1.207 | 2.414 |
| 2 | 19.314 | 16 | 2.414 | 4.828 |
| 3 | 43.456 | 24 | 3.621 | 7.243 |
| 5 | 120.711 | 40 | 6.036 | 12.071 |
| 7 | 236.589 | 56 | 8.050 | 16.899 |
| 10 | 482.843 | 80 | 12.071 | 24.142 |
| 12 | 694.493 | 96 | 14.485 | 28.971 |
Why the Constant is 2(1+√2)
A regular octagon can be constructed by taking a square with side (s√2 + s) and cutting the four corners off. The area equals the big square area minus the four corner triangles. Each corner is an isosceles right triangle with legs of length s/√2. When you work through the algebra, the result simplifies to 2(1+√2)×s². The constant 2(1+√2) ≈ 4.8284 is fixed for all regular octagons regardless of size.
Real-World Applications of Octagon Calculations
- Stop signs: Standard US stop signs are regular octagons with 12-inch sides — area ≈ 703 in²
- Gazebos and pavilions: Octagonal platforms are common in garden structures; area formula gives flooring material needed
- Bolt heads: Many metric bolt heads are regular octagons; width across flats determines the wrench size needed
- Architecture: Octagonal towers, fountains, and floor plans all require area and perimeter calculations
- Tiles: Octagonal ceramic tiles need area calculations for coverage estimates
💡 Quick Approximation: Area ≈ 4.83 × s². This is easy to remember and gives less than 0.01% error for any size octagon. For a stop sign with 30 cm sides: Area ≈ 4.83 × 900 = 4,347 cm². The exact answer is 4,328 cm² — less than 0.5% difference.
Frequently Asked Questions
Area = 2(1+√2) × s² where s is the side length. The constant 2(1+√2) ≈ 4.8284 is fixed for all regular octagons. For example, side = 5: Area = 4.8284 × 25 = 120.71 sq units. This formula works because a regular octagon can be decomposed into a central rectangle and triangles whose total area simplifies to this expression.
Multiply the square of the side length by 2(1+√2) ≈ 4.8284. For side = 4: Area = 4.8284 × 16 = 77.25 sq units. You can also use Area = 2s² + 2√2 × s², which is the same formula expanded. Both give the same result.
First convert circumradius R to side length: s = R × 2sin(π/8) ≈ R × 0.7654. Then apply Area = 2(1+√2) × s². Alternatively, Area = 2√2 × R² directly. For R = 6: Area = 2 × 1.4142 × 36 = 101.82 sq units.
The apothem is the perpendicular distance from the center to the middle of any side: apothem = s × (1+√2)/2 ≈ s × 1.2071. For s = 5, apothem = 6.036. The area can also be computed as Area = ½ × perimeter × apothem = ½ × 8s × apothem = 4s × apothem.
Perimeter = 8 × s, since all 8 sides are equal. For side = 5, perimeter = 40 units. For side = 12, perimeter = 96 units. This is the simplest of the octagon formulas since it is just the side length multiplied by 8.
Each interior angle of a regular octagon is 135°. This comes from (n−2)×180°/n = (8−2)×180/8 = 6×180/8 = 135°. The sum of all 8 interior angles is 8×135° = 1,080°. The exterior angle at each vertex is 180°−135° = 45°, and all 8 exterior angles sum to 360°.
A regular octagon has all 8 sides equal and all 8 angles equal at 135°. An irregular octagon has 8 sides but they may differ in length and the angles may be unequal. The formula Area = 2(1+√2)s² only works for regular octagons. For irregular octagons, divide the shape into triangles, compute each triangle’s area separately, and sum them.
Width across flats (between two parallel sides) = s × (1+√2) ≈ s × 2.4142. For s = 5, width = 12.071 units. Width across corners (vertex to vertex) = s × √(4+2√2) ≈ s × 2.6131. For s = 5, diagonal = 13.066 units. The corner-to-corner width is always greater than the flat-to-flat width.
Stop signs are the most familiar regular octagons — US standard stop signs have 12-inch sides. Other examples include octagonal gazebos, some clock faces, bolt heads, octagonal floor tiles, and towers in medieval castles. The octagon is popular in architecture because it approximates a circle while being constructable with straight walls.
Use the approximation Area ≈ 4.83 × s². This is easy to compute mentally and gives less than 0.1% error. For example, side = 10: Area ≈ 4.83 × 100 = 483 sq units. The exact answer is 482.84, so the approximation is off by less than 0.2 sq units. For rough estimates, simply multiply s² by 5 (slightly over-estimates by about 3.5%).
Related Calculators
Popular Calculators
Aspect Ratio Calculator
Math
Fraction to Percent Calculator
Math
Golf Altitude Calculator
Math
Hours to Time Calculator
Math
Stripe Fee Calculator
Finance
Pain & Suffering Calculator
Legal
TDEE Calculator
Health
Roofing Calculator
Construction
GPA Calculator
Education
Mortgage Refinance Calculator
Finance
CPM Calculator
Marketing
Markup Calculator
Finance