What is the octagon’s area?
Type any one measure of a regular octagon, such as its side, area or width across flats. The octagon calculator finds the side and then every other measure from it.
- Area (A)
- 120.711 in²
A regular octagon with side 5 in has an area of 120.711 in² and a perimeter of 40 in.
- Perimeter (P)
- 40 in
- Apothem (inradius)
- 6.036 in
- Circumradius (R)
- 6.533 in
- Width across flats
- 12.07 in
- Long diagonal (D)
- 13.07 in
- Short diagonal
- 9.239 in
Area (A): 120.711 in². A regular octagon with side 5 in has an area of 120.711 in² and a perimeter of 40 in.
How to calculate
Finds every measure of a regular octagon from any one of them: side, perimeter, area, apothem, circumradius, width and the long and short diagonals.
Example with the default inputs (Side (s) 5 in): A regular octagon with side 5 in has an area of 120.711 in² and a perimeter of 40 in.
Formula: P = 8s; apothem = s(1 + √2) ÷ 2; A = ½ × apothem × P = 2(1 + √2)s²; R = s√(4 + 2√2) ÷ 2; width = s(1 + √2); long diagonal = s√(4 + 2√2); short diagonal = s√(2 + √2).
- The octagon is regular: eight equal sides and eight equal angles of 135°.
- Type one measure; the side follows from it, and every other measure from the side.
Worked examples
Each example is checked against the calculator on every build.
- Side (s) 196.9 in gives Perimeter (P) 1,575 in, Apothem (inradius) 237.6 in, Area (A) 187,102 in², Circumradius (R) 257.2 in, Width across flats 475.2 in, Long diagonal (D) 514.4 in, Short diagonal 363.7 in.Source: OpenStax, Contemporary Mathematics, §10.6 Area (regular polygon A = ½ap, with a the apothem and p the perimeter). https://openstax.org/books/contemporary-mathematics/pages/10-6-area
- Perimeter (P) 629.9 in gives Side (s) 78.74 in, Area (A) 29,936.3 in², Width across flats 190.1 in.Source: OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; the angles of an n-gon add to (n − 2) × 180°). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference; OpenStax, Contemporary Mathematics, §10.6 Area (regular polygon A = ½ap, with a the apothem and p the perimeter). https://openstax.org/books/contemporary-mathematics/pages/10-6-area
- Area (A) 155,000 in² gives Side (s) 179.2 in, Perimeter (P) 1,433 in.Source: OpenStax, Contemporary Mathematics, §10.6 Area (regular polygon A = ½ap, with a the apothem and p the perimeter). https://openstax.org/books/contemporary-mathematics/pages/10-6-area
- Width across flats 393.7 in gives Side (s) 163.1 in, Apothem (inradius) 196.9 in.Source: OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; the angles of an n-gon add to (n − 2) × 180°). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference
How it works
A regular octagon has eight equal sides s and eight angles of 135°. Each outside angle is 45°, so the octagon is a square of width s(1 + √2) with four right triangles (legs s ÷ √2) cut from its corners. From that:
- Perimeter: P = 8s.
- Width across flats (between opposite sides, also the middle diagonal): w = s(1 + √2).
- Apothem (center to the middle of a side): ap = w ÷ 2 = s(1 + √2) ÷ 2.
- Area: A = ½ × ap × P = 2(1 + √2)s².
- Circumradius (center to a corner): R = s ÷ (2 sin 22.5°) = s√(4 + 2√2) ÷ 2.
- Long diagonal (opposite corners, through the center): D = 2R = s√(4 + 2√2).
- Short diagonal (two corners with one between them): s√(2 + √2).
Type one measure. The page finds the side from it (s = P ÷ 8, s = 2ap ÷ (1 + √2), s = √(A ÷ (2(1 + √2))), s = 2R ÷ √(4 + 2√2), s = w ÷ (1 + √2), s = D ÷ √(4 + 2√2) or s = short diagonal ÷ √(2 + √2)), then every other measure from the side. Two typed measures that do not fit one octagon give no answer.
Rules
- Every measure must be more than 0.
- The side can be at most 10⁹ m, the perimeter 8 × 10⁹ m, the area 5 × 10¹⁸ m², the apothem, circumradius and short diagonal 2 × 10⁹ m, and the width and long diagonal 3 × 10⁹ m. A value worked out past its limit gives no answer.
Output format. Values are double-precision numbers in the unit shown next to each field. √2 is the double-precision square root of 2.
Sources are OpenStax Contemporary Mathematics, because Algebra and Trigonometry 2e has no section on regular polygons.
Worked examples by hand
Side 5. P = 8 × 5 = 40. ap = 5(1 + √2) ÷ 2 = 6.0355. A = ½ × 6.0355 × 40 = 50(1 + √2) = 120.7107. R = 5√(4 + 2√2) ÷ 2 = 6.5328. w = 5(1 + √2) = 12.0711. D = 13.0656. Short diagonal 5√(2 + √2) = 9.2388.
Perimeter 16. s = 16 ÷ 8 = 2. A = 2(1 + √2) × 4 = 19.3137. w = 2(1 + √2) = 4.8284.
Area 100. s = √(100 ÷ 4.8284) = √20.7107 = 4.5509. P = 8 × 4.5509 = 36.4072.
Width 10. s = 10 ÷ (1 + √2) = 10(√2 − 1) = 4.1421. ap = 10 ÷ 2 = 5.
Other questions people ask
How do I find the area of a regular octagon?
Use A = 2(1 + √2)s², or half the apothem times the perimeter. An octagon with 5 cm sides has area 2(1 + √2) × 25 = 50(1 + √2) ≈ 120.71 cm².
What is the apothem of an octagon?
The distance from the center to the middle of a side, at a right angle to the side. In a regular octagon it is s(1 + √2) ÷ 2, about 1.2071 times the side.
How do I find the side from the width of an octagon?
The width across flats (between two opposite sides) is s(1 + √2), so s = width ÷ (1 + √2) = width × (√2 − 1). A 10 in wide octagon, such as a stop sign shape, has sides of about 4.142 in.
How do I find the side from the area?
Rearrange A = 2(1 + √2)s²: s = √(A ÷ (2(1 + √2))). An area of 100 in² gives a side of about 4.551 in.
What are the diagonals of a regular octagon?
There are three lengths. The short diagonal skips one corner and is s√(2 + √2). The middle one joins corners three apart and equals the width, s(1 + √2). The long diagonal joins opposite corners through the center and is s√(4 + 2√2), twice the circumradius.
What is each angle inside a regular octagon?
Each inside angle is 135°. The angles of an eight-sided polygon add up to (8 − 2) × 180° = 1,080°, and 1,080° ÷ 8 = 135°.