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Regular Octagon

A regular octagon has 8 equal sides and 8 interior angles of 135ยฐ. Area = 2(1+โˆš2)sยฒ โ‰ˆ 4.828sยฒ. The stop sign is the most recognizable octagon.

Concept Fundamentals
A = 2(1+โˆš2)sยฒ
Area
P = 8s
Perimeter
135ยฐ
Interior โˆ 
r โ‰ˆ 1.207s
Inradius

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Stop signs use octagons for visibility from any approach angle. Area is larger than a square of the same side length. Only one measurement needed to derive all other properties.

Key quantities
A = 2(1+โˆš2)sยฒ
Area
Key relation
P = 8s
Perimeter
Key relation
135ยฐ
Interior โˆ 
Key relation
r โ‰ˆ 1.207s
Inradius
Key relation

Ready to run the numbers?

Why: Octagons appear in stop signs (visibility from any angle), gazebos, and architectural elements. The 135ยฐ angles provide good structural balance.

How: Area = 2(1+โˆš2)sยฒ โ‰ˆ 4.828sยฒ. Inradius โ‰ˆ 1.207s, circumradius โ‰ˆ 1.307s. Height = 2 ร— inradius.

Stop signs use octagons for visibility from any approach angle.Area is larger than a square of the same side length.
Sources:Wolfram MathWorld - OctagonKhan Academy - Geometry

Run the calculator when you are ready.

Octagon CalculatorEnter side, area, perimeter, inradius, or circumradius to compute all properties
๐Ÿ›‘
GEOMETRY2D Shapes

Octagon โ€” The Stop Sign Shape

Calculate any property from just one measurement. 5 input modes: side length, area, perimeter, inradius, or circumradius.

Hexagon โ†’Polygon โ†’Square โ†’

๐Ÿ“ Real-World Examples โ€” Click to Load

Calculation Settings

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿงฎ Fascinating Math Facts

๐Ÿ›‘

Octagon area = 2(1+โˆš2)sยฒ โ€” the stop sign shape.

โ€” Formula

135ยฐ

Each interior angle of a regular octagon is 135ยฐ.

โ€” Property

๐Ÿ“‹ Key Takeaways

  • โ€ข A regular octagon has 8 equal sides and 8 interior angles of 135ยฐ each
  • โ€ข The stop sign is the most recognizable octagon โ€” chosen for visibility from any angle
  • โ€ข Area = 2(1 + โˆš2)sยฒ โ‰ˆ 4.828 ร— sยฒ โ€” larger than a square of the same side
  • โ€ข Inradius (apothem) โ‰ˆ 1.207 ร— side; Circumradius โ‰ˆ 1.307 ร— side
  • โ€ข Only one measurement (side, area, perimeter, inradius, or circumradius) is needed to calculate everything

๐Ÿ’ก Did You Know?

๐Ÿ›‘Stop signs were chosen as octagons because the shape is instantly recognizable from any angle โ€” even upside down or partially obscuredSource: MUTCD Traffic Standards
๐ŸฅŠThe UFC Octagon is the iconic fighting arena โ€” 30 feet across (flat-to-flat), designed for visibility and camera anglesSource: UFC History
๐Ÿ›๏ธThe Dome of the Rock in Jerusalem and the Baptistery of Florence use octagonal bases โ€” a symbol of regeneration in Roman and Islamic architectureSource: Architectural History
๐Ÿช™The British 50p is a heptagon (7 sides), but many coins use octagonal or near-octagonal shapes for easier vending machine recognitionSource: Royal Mint
๐Ÿ“An octagon has 8 lines of symmetry and rotational symmetry of order 8 โ€” rotate 45ยฐ and it looks identicalSource: Wolfram MathWorld
๐Ÿ Octagonal rooms were popular in Victorian and Gothic Revival architecture โ€” they maximize floor space while creating interesting sightlinesSource: Architectural History

๐Ÿ“– Octagon Formulas Explained

Area of a Regular Octagon

The area measures the space enclosed within the octagon. Divide the octagon into 8 isosceles triangles from the center and sum their areas.

A=2(1+2)s2โ‰ˆ4.828ร—s2A = 2(1 + \sqrt{2}) s^2 \approx 4.828 \times s^2

where s = side length

Example: s = 5 cm โ†’ A โ‰ˆ 120.7 cmยฒ. An octagon encloses more area than a square of the same side.

Perimeter of a Regular Octagon

The perimeter is the total distance around the octagon โ€” 8 equal sides.

P=8sP = 8s

Example: s = 5 cm โ†’ P = 40 cm.

Inradius and Circumradius

Inradius (apothem) = distance from center to side. Circumradius = distance from center to vertex.

r=s(1+2)2โ‰ˆ1.207sr = \frac{s(1 + \sqrt{2})}{2} \approx 1.207s
R=s2sinโก(ฯ€/8)โ‰ˆ1.307sR = \frac{s}{2\sin(\pi/8)} \approx 1.307s

Height (flat-to-flat) = 2r.

Inverse Formulas โ€” Find Side from Any Measurement

s=A/2(1+2)s = \sqrt{A / 2(1+\sqrt{2})}

From Area

s=P8s = \frac{P}{8}

From Perimeter

sโ‰ˆr1.207s \approx \frac{r}{1.207}

From Inradius

sโ‰ˆR1.307s \approx \frac{R}{1.307}

From Circumradius

s=h2.414s = \frac{h}{2.414}

From Height

๐ŸŽฏ Expert Tips

๐Ÿ’ก Stop Sign Design

US stop signs are 30 inches across the flats (inradius ร— 2). If designing signage, use inradius mode for the flat-to-flat dimension.

๐Ÿ’ก Gazebo & Floor Plans

For octagonal floor plans, measure the flat-to-flat distance (height) โ€” that's 2 ร— inradius. Enter half that value as inradius.

๐Ÿ’ก Area Scaling Rule

Area scales by the square of the linear factor: double the side โ†’ 4ร— the area. Triple the side โ†’ 9ร— the area.

๐Ÿ’ก Circumscribed Circle

The octagon encloses about 90% of the area of its circumscribed circle โ€” more efficient than a square (~63.7%).

โš–๏ธ Why Use This Calculator vs. Other Tools?

FeatureThis CalculatorBasic CalculatorManual
Multiple input modes (side/area/P/r/R)โœ…โŒโŒ
Step-by-step solutionsโœ…โŒโœ…
Interactive chartsโœ…โŒโŒ
Copy & share resultsโœ…โŒโŒ
All properties at onceโœ…โš ๏ธ Limitedโš ๏ธ Tedious
Unit support (mm to yd)โœ…โš ๏ธ LimitedโŒ
Real-world examples (stop sign, etc.)โœ…โŒโŒ
Mobile-friendlyโœ…โš ๏ธ VariesโŒ

๐Ÿ“Š Octagon Quick Facts

8
Equal Sides
135ยฐ
Interior Angle
8
Lines of Symmetry
4.828
Area Factor (ร—sยฒ)

โ“ Frequently Asked Questions

How do I find the area of an octagon?

Use A = 2(1 + โˆš2)sยฒ โ‰ˆ 4.828sยฒ where s is the side length. For example, s = 5 cm โ†’ A โ‰ˆ 120.7 cmยฒ. If you know the perimeter instead, divide by 8 to get the side first.

What is the difference between a regular and irregular octagon?

A regular octagon has all 8 sides equal and all interior angles 135ยฐ. An irregular octagon has 8 sides but they may vary in length and angle. This calculator handles only regular octagons.

Why are stop signs octagonal?

The octagon was chosen for stop signs because it is highly recognizable from any angle โ€” even upside down or partially obscured. The shape signals "stop" before you can read the text.

What is the inradius of an octagon?

The inradius (apothem) is the perpendicular distance from the center to any side. For a regular octagon, r โ‰ˆ 1.207 ร— side length. The height (flat-to-flat) equals 2 ร— inradius.

How does the octagon compare to a square of the same side?

An octagon encloses more area: A_octagon โ‰ˆ 4.828sยฒ vs A_square = sยฒ. So the octagon has about 4.83ร— the area of a square with the same side length.

What is the interior angle sum of an octagon?

The sum of interior angles in any octagon is (8-2)ร—180ยฐ = 1080ยฐ. Each interior angle of a regular octagon is 1080ยฐ/8 = 135ยฐ.

How do I find the side length from the area?

Use s = โˆš(A / 2(1+โˆš2)) โ‰ˆ โˆš(A / 4.828). For example, if A = 100 cmยฒ, then s โ‰ˆ โˆš(100/4.828) โ‰ˆ 4.55 cm.

What is the UFC Octagon size?

The UFC Octagon is 30 feet across (flat-to-flat), so the inradius is 15 feet. Side length โ‰ˆ 15/1.207 โ‰ˆ 12.4 feet. Perimeter โ‰ˆ 99 feet.

โš ๏ธ Disclaimer: This calculator provides mathematically precise results based on standard Euclidean geometry formulas for a perfect regular octagon. Real-world measurements (stop signs, gazebos, coins) may vary due to manufacturing tolerances, measurement precision, and material properties. The British 50p is a heptagon, not an octagon โ€” used here for educational comparison. Always verify measurements for construction and engineering projects.

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