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Regular Octagon Area Calculator

Find the area of a regular octagon.

Result

77.25 units²
32.00 units

How it is calculated

For a regular octagon, with eight equal sides and equal angles, the area is A = 2 × (1 + √2) × s² and the perimeter is P = 8 × s, where s is the length of one side. The factor 2 × (1 + √2) is about 4.828. It comes from a square of side s(1 + √2) with four small right-angled corners cut off.

Use any length unit; the output says 'units' and 'units²', meaning the unit you entered and its square. Results are rounded to two decimals and the side must be greater than zero. If you know the distance across the flats rather than the side, divide it by 1 + √2, about 2.414, to get the side first.

Worked example

A regular octagon with side 4. Square the side: 4² = 16. The factor is 2 × (1 + 1.4142) = 4.8284. Multiply: 16 × 4.8284 = 77.25 units². Perimeter: 8 × 4 = 32.00 units. For a stop-sign style shape measured 4 units per side, that is the area of the face.

Try this example →

Common mistakes

  • Entering the width across the flats as the side. For a regular octagon that width is about 2.414 times the side, so the area would be far too large.
  • Entering the corner-to-corner width. Divide it by about 2.613 first to get the side, otherwise the result is wrong.
  • Using the formula for an irregular eight-sided shape. All eight sides and angles must be equal.
  • Reading the output unit label literally. 'units' means whatever you typed, so centimetres give cm² for the area.
  • Confusing area with perimeter when ordering material. Edging or trim uses the perimeter, while paint, tile or fabric uses the area.

About this calculator

This calculator finds the area, A = 2(1+√2)·s², and perimeter, P = 8·s, of a regular octagon — an eight-sided polygon with equal sides and equal interior angles of 135° — from its side length s alone. The area formula comes from splitting the octagon into eight congruent isosceles triangles meeting at the center, or equivalently from cutting the corners off a square.

Architects and builders use this when designing octagonal rooms, gazebos, or turrets, since the shape is popular for its wide field of view and structural stability; sign manufacturers use the same geometry for stop signs (arguably the most recognizable regular octagon in daily life). Woodworkers and DIY builders constructing octagonal tables or planters, and geometry students verifying textbook problems, use this calculator to get exact area and perimeter figures from a single side measurement.

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Comments (1)

  • Editorial team

    A handy cross-check: a regular octagon covers about 82.8% of the square drawn around it, so area ≈ 0.828 × W², where W is the width across the flats. With a side of 4, W is about 9.66 and 0.828 × 9.66² lands on the 77.25 the tool shows. If your hand estimate disagrees badly, the side or the width was likely mixed up. What are you building, a gazebo, a table or a planter?

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