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Octagon Calculator

Calculate the area, perimeter, and other properties of a regular octagon from its side length.

What this tool does

Calculate the area, perimeter, and other properties of a regular octagon from its side length. You fill in these fields: Side length. It shows area, perimeter, width across flats, apothem (inradius), circumradius, interior angle, and central angle.

Inputs

  • Side length (units): A number. Must be greater than 0. Length of one side. Use the same unit for every measurement.

How to use

  1. Enter the side length in units.
  2. Read the result: area, perimeter, width across flats, apothem (inradius), circumradius, interior angle, and central angle. It appears once every required field has a valid value.

Worked examples

Example 1. Inputs:

  • Side length: 5 units

Results:

  • Area: 120.710678 square units (A = 2(1 + √2) × s²)
  • Perimeter: 40 units (P = 8 × s)
  • Width across flats: 12.071068 units (Distance between opposite sides, s(1 + √2))
  • Apothem (inradius): 6.035534 units (Distance from center to side midpoint)
  • Circumradius: 6.532815 units (Distance from center to vertex)
  • Interior angle: 135° (Angle at each vertex: (8 - 2) × 180 ÷ 8)
  • Central angle: 45° (Angle from center to adjacent vertices: 360 ÷ 8)
  • Note: This calculator is for regular octagons only, with all sides and all angles equal.

How this is worked out: Area = 2(1 + √2) × s² = 2 × 2.414213562 × 25 = 120.7106781. Perimeter = 8 × 5 = 40. Apothem = s(1 + √2) / 2 = 5 × 2.414213562 / 2 = 6.035533906. Circumradius = s / (2 sin(π/8)) = 5 / (2 × 0.3826834324) = 6.532814824. Interior angle = (8 - 2) × 180 / 8 = 135. Central angle = 360 / 8 = 45.

Example 2. Inputs:

  • Side length: 3 units

Results:

  • Area: 43.455844 square units (A = 2(1 + √2) × s²)
  • Perimeter: 24 units (P = 8 × s)
  • Width across flats: 7.242641 units (Distance between opposite sides, s(1 + √2))
  • Apothem (inradius): 3.62132 units (Distance from center to side midpoint)
  • Circumradius: 3.919689 units (Distance from center to vertex)
  • Interior angle: 135° (Angle at each vertex: (8 - 2) × 180 ÷ 8)
  • Central angle: 45° (Angle from center to adjacent vertices: 360 ÷ 8)
  • Note: This calculator is for regular octagons only, with all sides and all angles equal.

How this is worked out: Area = 2 × 2.414213562 × 9 = 4.828427125 × 9 = 43.45584412. Perimeter = 8 × 3 = 24. Apothem = 3 × 2.414213562 / 2 = 3.621320343. Circumradius = 3 / (2 × 0.3826834324) = 3 × 1.306562965 = 3.919688895. Angles are 135 and 45 for any side length.

Example 3. Inputs:

  • Side length: 1 unit

Results:

  • Area: 4.828427 square units (A = 2(1 + √2) × s²)
  • Perimeter: 8 units (P = 8 × s)
  • Width across flats: 2.414214 units (Distance between opposite sides, s(1 + √2))
  • Apothem (inradius): 1.207107 units (Distance from center to side midpoint)
  • Circumradius: 1.306563 units (Distance from center to vertex)
  • Interior angle: 135° (Angle at each vertex: (8 - 2) × 180 ÷ 8)
  • Central angle: 45° (Angle from center to adjacent vertices: 360 ÷ 8)
  • Note: This calculator is for regular octagons only, with all sides and all angles equal.

How this is worked out: Area = 2(1 + √2) × 1 = 2 × 2.414213562 = 4.828427125. Perimeter = 8 × 1 = 8. Apothem = 2.414213562 / 2 = 1.207106781. Circumradius = 1 / (2 × 0.3826834324) = 1.306562965. Angles are 135 and 45.

Limitations

  • Only regular octagons, with all sides and angles equal, are calculated.
  • The side length must be greater than zero.
  • Each result is a single figure from one side length. The tool does not calculate cutting, material or waste amounts.

FAQs

Q: Does this work for octagons that are not regular?
A: No. The results assume all eight sides are equal and all eight angles are equal. For an irregular octagon the area and the other figures will be wrong.
Q: What is the width across flats?
A: It is the distance between two opposite sides, measured straight across the octagon. It equals the side length times (1 + √2), and it is twice the apothem.
Q: Which unit should I enter?
A: Any length unit works, such as feet or metres. Enter the side length in that unit. The perimeter and distances come out in the same unit, and the area comes out in that unit squared.
Q: Why are the angles always the same?
A: Every regular octagon has the same angles. Each interior angle is 135 degrees and each central angle is 45 degrees, whatever the side length.

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