Math
Octagon Calculator
Calculate side, perimeter, area, apothem, diagonals, and angles of a regular octagon from any one measurement.
Regular octagon dimensions
Pick which dimension you have. The rest are derived.
Area
482.8427
- Side (s)
- 10
- Perimeter
- 80
- Apothem
- 12.0711
- Long diagonal
- 26.1313
- Short diagonal
- 24.1421
- Circumradius
- 13.0656
- Inradius
- 12.0711
- Interior angle
- 135 deg
- Exterior angle
- 45 deg
Area formula
A = 2 (1 + sqrt(2)) s^2
Apothem formula
a = (s / 2) (1 + sqrt(2))
Long diagonal formula
d_long = s sqrt(4 + 2 sqrt(2))
Frequently Asked Questions about the Octagon Calculator
What is the area formula for a regular octagon?
A regular octagon with side length s has area A = 2 (1 + sqrt(2)) s^2, which equals about 4.828427 times s squared. For s = 10, the area is about 482.84 square units.
What is the difference between the long and short diagonal?
The long diagonal connects two opposite vertices and equals s sqrt(4 + 2 sqrt(2)), about 2.613126 s. The short diagonal connects two opposite flat sides and equals s (1 + sqrt(2)), about 2.414214 s, the same as twice the apothem.
How do I find the side length from the area?
Rearrange the area formula: s = sqrt(A / (2 (1 + sqrt(2)))) = sqrt(A / 4.828427). For example, an area of 100 square units gives a side of about 4.553 units.
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, apothem a = (s / 2) (1 + sqrt(2)), about 1.207107 s. It is also the inradius (radius of the inscribed circle).
What are the interior and exterior angles of a regular octagon?
Every interior angle measures 135 degrees, and every exterior angle measures 45 degrees. The eight interior angles add up to (8 - 2) x 180 = 1080 degrees.