Regular Octagon Calculator

Enter the side length to calculate area, perimeter, apothem, diagonal, and interior angles of a regular octagon.

Enter Side Length

Result

Area
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square units

Step-by-Step Solution

What Is a Regular Octagon?

A regular octagon is an eight-sided polygon where all sides have equal length and all interior angles are equal. Each interior angle measures exactly 135 degrees, and the sum of all interior angles is 1080 degrees. The regular octagon is one of the most recognizable polygons, commonly seen in stop signs and architectural designs.

Octagon Formulas

Area

The area of a regular octagon with side length s.

A = 2(1 + sqrt(2)) x s²

Perimeter

The total length of all eight sides.

P = 8s

Apothem

The distance from center to the midpoint of a side.

a = (s/2)(1 + sqrt(2))

Long Diagonal

The longest diagonal passing through the center.

d = s x sqrt(4 + 2sqrt(2))

Properties of a Regular Octagon

  • 8 equal sides and 8 equal angles.
  • Each interior angle is 135 degrees.
  • Each exterior angle is 45 degrees.
  • It has 20 diagonals.
  • It has 8 lines of symmetry and rotational symmetry of order 8.
  • It can tessellate the plane when combined with squares.

Real-World Applications

The octagon shape is widely used in everyday life. Stop signs worldwide are regular octagons, chosen for their distinctive shape that is easily recognizable even from behind. In architecture, octagonal floor plans appear in towers, baptisteries, and gazebos. The shape also appears in flooring tile patterns and decorative designs.

Derivation of the Area Formula

The area formula A = 2(1 + sqrt(2))s² can be derived by dividing the octagon into 8 isosceles triangles from the center. Each triangle has a base of length s and a height equal to the apothem. The area of each triangle is (1/2) x s x apothem, and multiplying by 8 gives the total area.