AJ Designer

Octagon Calculator (Regular)

Area equals two times one plus root two, times s squared

Solution

—
Share:

Calculate Regular Octagon Area from Side

Use this form when the side length of a regular octagon (all eight sides equal, all interior angles 135°) is known. The coefficient 2(1+√2) ≈ 4.828 comes from decomposing the octagon into 8 isosceles triangles meeting at the center.

A = 2(1 + √2) s²

Calculate Octagon Perimeter from Side

Use this form to compute the perimeter when the side length is known — fencing, trim, or edge banding for octagonal layouts like gazebos and stop signs.

P = 8s

Calculate Octagon Apothem from Side

Use this form for the apothem — the perpendicular distance from the center to the midpoint of any edge. This is also the inscribed-circle radius and equals half the short (across-flats) diagonal.

a = (s/2)(1 + √2)

Calculate Octagon Side from Area

Use this rearrangement when the area is known and you need the side length — e.g., sizing a stop sign or octagonal patio to a target footprint.

s = √(A / (2(1 + √2)))

How It Works

This regular octagon calculator solves A = 2(1+√2)s² for area, P = 8s for perimeter, and a = (s/2)(1+√2) for the apothem (the center-to-edge-midpoint distance, also the inscribed-circle radius). The inverse s = √(A / (2(1+√2))) recovers the side from the area. A regular octagon has eight equal sides, eight equal 135° interior angles, and two distinct diagonals: the long diagonal (vertex to opposite vertex) is s · √(4+2√2) ≈ 2.613s, and the short diagonal (vertex skipping one vertex) is s(1+√2) ≈ 2.414s — exactly twice the apothem.

Example Problem

A regular octagonal stop sign has 4 in sides. Compute its area, perimeter, apothem, and diagonals.

  1. Knowns: s = 4 in
  2. Area: A = 2(1 + √2) · s² = 2(1+√2) · 16 = 32(1+√2) ≈ 77.255 in²
  3. Perimeter: P = 8s = 8 · 4 = 32 in
  4. Apothem (center to edge midpoint): a = (s/2)(1+√2) = 2(1+√2) ≈ 4.828 in
  5. Circumradius (center to vertex): R = (s/2)√(4+2√2) ≈ 5.226 in
  6. Long diagonal: s·√(4+2√2) ≈ 10.453 in; Short diagonal: s(1+√2) ≈ 9.657 in

U.S. road stop signs are regular octagons. Standard MUTCD spec is 30 in across-flats (= short diagonal), which corresponds to s ≈ 30 / (1+√2) ≈ 12.43 in per side.

When to Use Each Variable

  • Solve for Area — when the side length is known — stop signs, octagonal patios, gazebos, decorative tiles.
  • Solve for Perimeter — when you need the boundary length — fencing for an octagonal gazebo, trim around an octagonal window.
  • Solve for Apothem — when you need the inscribed-circle radius or the across-flats half-distance — tooling, machining, sign-mounting layouts.
  • Solve for Side — when the area is known and you need the side length — sizing a sign or patio to a target footprint.

Key Concepts

A regular octagon has eight equal sides and eight equal interior angles of 135° (since the interior-angle sum of any n-gon is (n−2)·180°, here 6·180° = 1080°, divided by 8). The shape sits between the hexagon and the circle: as the number of sides of a regular polygon grows, the figure approaches its circumscribed circle, and the octagon already feels noticeably circular. Octagons do NOT tile the plane by themselves — they leave square gaps — which is why a stop sign placed on a square mounting plate is so natural a pairing.

Applications

  • Road signs: U.S. stop signs are regular octagons (MUTCD standard 30 in across-flats)
  • Architecture: octagonal towers, gazebos, cupolas, and bay windows
  • Tiling: octagon + square tile patterns cover floors without gaps (the square fills the gap an octagon-only tiling leaves)
  • Mechanical: octagonal sockets and bolt heads where extra wrenching surface is needed vs. a hex
  • Heraldry and decor: octagonal mirrors, picture frames, and clock faces

Common Mistakes

  • Confusing the apothem with the circumradius — apothem a = (s/2)(1+√2) ≈ 1.207s reaches edge midpoints, while circumradius R = (s/2)√(4+2√2) ≈ 1.307s reaches the vertices.
  • Forgetting that an octagon has 8 sides — perimeter is 8s, not 6s or 4s.
  • Confusing the regular octagon (all sides equal, all angles 135°) with irregular octagons (any 8-gon) — this calculator handles the regular case only.
  • Using a hexagon-style ‘circumradius equals side’ rule — that property is unique to hexagons. For octagons, R > s.
  • Mixing up the two diagonals: long (vertex to opposite vertex, ≈ 2.613s) vs. short (vertex skipping one, ≈ 2.414s). The short diagonal is the across-flats distance only when measured as 2·apothem, which is what stop-sign specs use.

Frequently Asked Questions

How do you calculate the area of a regular octagon?

Use A = 2(1 + √2) · s², where s is the side length. The coefficient 2(1+√2) ≈ 4.828. For s = 4 in, A = 32(1+√2) ≈ 77.255 in².

What is the formula for the perimeter of an octagon?

For a regular octagon, P = 8s (eight equal sides times the side length). For s = 4 in, P = 32 in.

What is the apothem of a regular octagon?

The apothem is the perpendicular distance from the center to the midpoint of any side. For a regular octagon, a = (s/2)(1+√2) ≈ 1.207s. For s = 4 in, a ≈ 4.828 in.

What is the interior angle of a regular octagon?

135°. The interior-angle sum of any n-gon is (n − 2) · 180°; for n = 8 that is 6 · 180° = 1080°, divided by 8 vertices gives 135° per corner.

How big is a U.S. stop sign?

Federal MUTCD spec is 30 in across-flats (the short diagonal, = 2·apothem). That implies side s = 30 / (1+√2) ≈ 12.43 in and area ≈ 746 in² (≈ 5.18 ft²).

How do you find the side of an octagon given the area?

Rearrange A = 2(1+√2)s² to s = √(A / (2(1+√2))). For A = 77.255 in², s = √(77.255 / 4.828) = √16 = 4 in.

What are the diagonals of a regular octagon?

Two distinct diagonals: the LONG diagonal (vertex to opposite vertex) is s · √(4+2√2) ≈ 2.613s, and the SHORT diagonal (vertex skipping one) is s(1+√2) ≈ 2.414s. The short diagonal also equals 2·apothem — the across-flats distance used on stop-sign specs.

Do regular octagons tile the plane?

No, not by themselves — at any vertex three 135° angles sum to 405°, which overshoots 360°, and two angles only fill 270°, which leaves a gap. Octagons can tile the plane only when paired with squares (the truncated-square tiling), which is why octagonal road tiles and patios usually include square inserts.

Reference: Weisstein, Eric W. "Regular Octagon." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/RegularOctagon.html

Worked Examples

Stop Sign

How big is a U.S. stop sign?

Federal MUTCD spec sets the across-flats distance (= short diagonal = 2·apothem) at 30 in. Compute the side length and area.

  • Across-flats = 30 in → apothem a = 15 in
  • Solve a = (s/2)(1+√2) for s: s = 2a / (1+√2) = 30 / (1+√2)
  • Rationalize: s = 30(√2 − 1) ≈ 30 · 0.4142 ≈ 12.43 in
  • Area: A = 2(1+√2) · s² ≈ 4.828 · 154.5 ≈ 745.6 in²

Side s ≈ 12.43 in, area ≈ 745.6 in² (≈ 5.18 ft²)

Larger highway stop signs are 36 in or 48 in across-flats; scale s and A proportionally.

Gazebo

How much decking does an octagonal gazebo with 6 ft sides need?

Compute the floor area of a regular octagonal gazebo with 6 ft sides to estimate decking material.

  • Knowns: s = 6 ft
  • Formula: A = 2(1 + √2) · s²
  • A = 2(1+√2) · 36 = 72(1+√2) ≈ 173.8 ft²
  • Perimeter (railing): P = 8s = 48 ft

Floor area ≈ 173.8 ft², railing perimeter = 48 ft

Add 5–10% waste to the decking estimate to cover the angled cuts at each 135° corner.

Inverse Solve

What side does a 100 cm² octagonal tile need?

A decorative tile must have exactly 100 cm² area. Find the required side length.

  • Knowns: A = 100 cm²
  • Formula: s = √(A / (2(1+√2)))
  • s = √(100 / 4.828) ≈ √20.71
  • s ≈ 4.551 cm

Side ≈ 4.55 cm

Useful for sizing tiles, mosaic blocks, or octagonal coasters to a target footprint.

Regular Octagon Formulas

A regular octagon has eight equal sides and eight 135° interior angles. A single dimension, the side length s, fully determines its geometry.

Area equals two times one plus root two, times s squaredPerimeter equals eight sApothem equals s over two, times one plus root twoSide equals the square root of A divided by two times one plus root two
Regular octagon with side s and apothem aas

Where:

  • A — area (m², ft², in²)
  • P — perimeter (m, ft, in) = 8s
  • s — side length (all sides equal)
  • a — apothem (center to edge midpoint) = (s/2)(1+√2)
  • R — circumradius (center to vertex) = (s/2)√(4+2√2) ≈ 1.307s
  • Long diagonal — vertex to opposite vertex = s · √(4+2√2) ≈ 2.613s = 2R
  • Short diagonal — vertex skipping one = s(1+√2) ≈ 2.414s = 2·apothem (across-flats)

Related Calculators

  • Hexagon Calculator — regular 6-gon — area, perimeter, apothem, and diagonals
  • Pentagon Calculator — regular 5-gon — area, perimeter, apothem, and diagonals
  • Square Calculator — compute area, perimeter, and diagonal for a square (paired with octagons in truncated-square tilings)
  • Circle Calculator — compute area and circumference — the octagon's inscribed circle has radius equal to its apothem
  • Area Converter — switch between m², ft², in², and other area units

Related Sites