Octagon Calculator

What is an Octagon?

An octagon is a polygon with 8 sides and 8 angles. The word "octagon" comes from the Greek words "okto" (eight) and "gonia" (angle). A regular octagon is a special octagon where all sides are equal in length and all interior angles are equal (135° each).

Octagon Angles

For any octagon, the sum of all interior angles is always 1080°. This is calculated using the formula:

Sum of angles = (n - 2) × 180°

For octagon: (8 - 2) × 180° = 1080°

In a regular octagon, each interior angle measures exactly:

Interior angle = 1080° ÷ 8 = 135°

Octagon Area Formula

The area of a regular octagon can be calculated using:

Area = 2 × (1 + √2) × a²

Area ≈ 4.828427 × a²

Where a is the length of one side.

Alternative formula:

Area = Perimeter × Apothem / 2

Understanding the Area Formula

The octagon can be divided into 8 isosceles triangles from the center:

  • Each triangle has a base equal to the side length
  • The height of each triangle is the apothem (inradius)
  • Area of one triangle = base × height / 2
  • Total area = 8 × (base × height / 2) = perimeter × apothem / 2

Octagon Perimeter Formula

Perimeter = 8 × a

Simply multiply the side length by 8.

Octagon Diagonals

A regular octagon has 20 diagonals of three different lengths:

1. Short Diagonal (s)

s = a × √(2 + √2)

s ≈ 1.8478 × a

Connects vertices with one vertex in between (e.g., from vertex 1 to vertex 3).

2. Medium Diagonal (m)

m = a × (1 + √2)

m ≈ 2.4142 × a

Connects vertices with two vertices in between. Also called the height of the octagon.

3. Long Diagonal (l)

l = a × √(4 + 2√2)

l ≈ 2.6131 × a

The longest diagonal, connecting opposite vertices through the center.

Circumradius and Inradius

Circumradius (R)

The radius of the circle that passes through all 8 vertices:

R = a / 2 × √(4 + 2√2)

R = l / 2 (half the long diagonal)

R ≈ 1.3066 × a

Inradius (r) - Apothem

The radius of the largest circle that fits inside the octagon:

r = a / 2 × (1 + √2)

r = m / 2 (half the medium diagonal)

r ≈ 1.2071 × a

How to Calculate Octagon Properties

From Side Length (a)

  • Perimeter = 8a
  • Area = 2(1 + √2)a²
  • Short diagonal = a√(2 + √2)
  • Medium diagonal = a(1 + √2)
  • Long diagonal = a√(4 + 2√2)
  • Circumradius = a√(4 + 2√2) / 2
  • Inradius = a(1 + √2) / 2

From Perimeter (P)

  • Side length = P / 8
  • Then use side length formulas above

From Area (A)

  • Side length = √(A / (2(1 + √2)))
  • Then use side length formulas above

How to Draw a Regular Octagon

Method 1: Using a Circle

  1. Draw a circle (use a compass, glass, or coin)
  2. Draw a diameter line through the center
  3. Draw another diameter perpendicular to the first (now you have 4 points)
  4. Bisect each of the 4 sections to get 8 equally spaced points
  5. Connect adjacent points with straight lines
  6. You now have a regular octagon!

Method 2: Truncating a Square

  1. Start with a square
  2. "Chop off" each corner at 45° angles
  3. Make sure the cuts are equal on all corners
  4. The result is a regular octagon

Octagon Examples

Example 1: Octagon with side = 5 cm

Given: Side length a = 5 cm

Calculate:

  • Perimeter = 8 × 5 = 40 cm
  • Area = 2(1 + √2) × 5² = 2(2.414) × 25 = 120.71 cm²
  • Short diagonal = 5 × √(2 + √2) = 5 × 1.8478 = 9.24 cm
  • Medium diagonal = 5 × (1 + √2) = 5 × 2.414 = 12.07 cm
  • Long diagonal = 5 × √(4 + 2√2) = 5 × 2.613 = 13.07 cm
  • Circumradius = 13.07 / 2 = 6.53 cm
  • Inradius = 12.07 / 2 = 6.04 cm

Example 2: Octagon with perimeter = 80 cm

Given: Perimeter = 80 cm

Calculate:

  • Side length = 80 / 8 = 10 cm
  • Area = 2(1 + √2) × 10² = 482.84 cm²
  • Long diagonal = 10 × 2.613 = 26.13 cm

Example 3: Octagon with area = 200 cm²

Given: Area = 200 cm²

Calculate:

  • Side length = √(200 / 4.828) = √41.42 = 6.44 cm
  • Perimeter = 8 × 6.44 = 51.50 cm
  • Medium diagonal = 6.44 × 2.414 = 15.54 cm

Octagons in Real Life

1. Stop Signs

The most common real-world octagon is the stop sign. In many countries, stop signs are red octagons with white letters. The unique shape makes them instantly recognizable, even from behind or when covered with snow.

2. Octagon House

The famous Octagon House in Washington D.C., also known as the Colonel John Tayloe III House, was built in an octagonal shape. This architectural marvel features triangular staircases, oval rooms, and is surrounded by ghost stories due to its historical significance.

3. Octagon Tiles

Octagon tiles are popular in flooring, especially when combined with square tiles. The shape allows for complete coverage of a floor with an attractive pattern, commonly seen in kitchens and bathrooms.

4. Camera Apertures

Many camera apertures use an octagonal shape. You can identify this by looking at bright lights in photos - if you see an 8-pointed star pattern (called diffraction spikes), the camera has an octagonal aperture!

5. Architecture and Design

  • Gazebos and pavilions
  • Clock towers
  • Windows and skylights
  • Umbrellas and parasols
  • Wrestling and MMA fighting rings

Octagon Quick Reference

Property Formula Multiplier
Sides - 8
Perimeter 8a 8 × a
Area 2(1 + √2)a² 4.828 × a²
Short Diagonal a√(2 + √2) 1.848 × a
Medium Diagonal a(1 + √2) 2.414 × a
Long Diagonal a√(4 + 2√2) 2.613 × a
Circumradius a√(4 + 2√2) / 2 1.307 × a
Inradius a(1 + √2) / 2 1.207 × a
Interior Angle 1080° / 8 135°
Total Diagonals - 20

Frequently Asked Questions

How many sides does an octagon have?

An octagon has 8 sides. The word "octagon" comes from Greek: "okto" means eight and "gonia" means angle.

What is the sum of angles in an octagon?

The sum of all interior angles in any octagon is 1080°. This is calculated using the formula (n - 2) × 180° where n = 8 sides. For a regular octagon, each angle is 135°.

How do I calculate the area of a regular octagon?

Use the formula: Area = 2 × (1 + √2) × a² where a is the side length. Alternatively, if you know the perimeter and apothem: Area = Perimeter × Apothem / 2.

What is a regular octagon?

A regular octagon is an octagon where all 8 sides are equal in length and all 8 interior angles are equal (135° each). It has perfect symmetry and is the most common type of octagon seen in everyday life (like stop signs).

How many diagonals does an octagon have?

An octagon has 20 diagonals in total. In a regular octagon, these come in three lengths: short, medium, and long diagonals. The formula for the number of diagonals in any polygon is n(n-3)/2, where n is the number of sides.

What is the difference between convex and concave octagons?

A convex octagon has all interior angles less than 180° (all corners point outward). A concave octagon has at least one interior angle greater than 180° (at least one corner points inward). A regular octagon is always convex.

Can I make an octagon from a square?

Yes! You can create a regular octagon by truncating a square - cutting off all four corners at 45° angles with equal cuts. This is one of the easiest ways to draw an octagon by hand.

💡 Did You Know?

The Pentagon (US Department of Defense headquarters) and the Octagon House both showcase how regular polygons can be used in architecture. The octagon shape is easier to construct than a circle while still providing good space efficiency and aesthetic appeal!