Octagon Calculator
Calculate area, perimeter, and apothem for regular octagons (like stop signs!)
Input Values
Results
Step-by-Step Solution
A = 2(1 + √2) × s²
A = 2(1 + 1.414214) × 10²
A = 4.828427 × 100.0000
A = 0.0000
a = s(1 + √2) / 2
a = 10(1 + √2) / 2
a = 0.0000
Perimeter: P = 8s = 0.0000
Long Diagonal: d = s(1 + √2) = 0.0000
Interior Angle: (8-2) × 180° / 8 = 135°
Understanding Octagons
What is an Octagon?
An octagon is an eight-sided polygon. A regular octagon has all eight sides of equal length and all eight interior angles equal to 135°. The most familiar example of a regular octagon is the stop sign used in traffic control worldwide.
Octagon Formulas
Area: A = 2(1 + √2) × s² ≈ 4.828s²
Perimeter: P = 8s
Apothem: a = s(1 + √2) / 2 ≈ 1.207s
Radius: r = s / (2sin(π/8)) ≈ 1.307s
Long Diagonal: d = s(1 + √2) ≈ 2.414s
Interior Angle: 135°
Exterior Angle: 45°
The Stop Sign Connection
Stop signs are octagonal because the eight-sided shape is distinctive and easily recognizable even when partially obscured or viewed from an angle. The shape was standardized in the 1920s, and the octagon's unique geometry makes it stand out from other traffic signs (circles, triangles, rectangles).
Octagons in Architecture and Design
- The Dome of the Rock in Jerusalem features an octagonal base
- Many umbrella designs use octagonal canopies
- Octagonal towers appear in medieval castles and fortifications
- Floor tiles and decorative patterns often feature octagons
- Some clocks and watches use octagonal faces
Frequently Asked Questions
Why are stop signs octagonal?
Stop signs are octagonal for immediate recognition. The eight-sided shape is unique among traffic signs and can be identified by shape alone, even when the text or color is obscured. This was standardized in 1915 to help drivers recognize the most critical warning sign.
Can octagons tessellate?
Regular octagons alone cannot tessellate a plane because their 135° interior angles don't divide evenly into 360°. However, octagons can tessellate when combined with squares (octagon-square tiling), which is commonly seen in bathroom floors and historic architecture.
How do you construct a regular octagon?
One method: Draw a square, then draw its diagonals. Mark equal distances from each corner along the sides. Connect these marks to "cut off" the corners at 45° angles, creating eight equal sides. Another method uses a compass to divide a circle into eight equal parts.
What is the √2 in the octagon formula?
The √2 appears because a regular octagon can be constructed by cutting the corners off a square at 45° angles. The relationship between the octagon's dimensions and the original square involves √2, which is also the diagonal-to-side ratio of a square.
How many diagonals does an octagon have?
An octagon has 20 diagonals, calculated using the formula n(n-3)/2 where n=8. These include short diagonals, medium diagonals, and long diagonals that pass through the center. The long diagonals have length s(1 + √2).
What is the interior angle sum?
Using the formula (n-2) × 180° with n=8, the sum of interior angles is (8-2) × 180° = 1080°. Since all angles are equal in a regular octagon, each angle is 1080° / 8 = 135°.
How is an octagon related to a square?
A regular octagon can be formed by truncating (cutting off) the corners of a square. If you cut each corner at a 45° angle, removing isosceles right triangles with legs equal to s/(1+√2), you create a regular octagon with side length s.
What is octagonal symmetry?
A regular octagon has 8 lines of symmetry (4 through opposite vertices, 4 through midpoints of opposite sides) and rotational symmetry of order 8, meaning it looks identical after rotation by 45°. This high degree of symmetry makes it aesthetically pleasing.