Octagon Calculator
Calculate the area, perimeter, apothem, circumradius, diagonals, and other properties of a regular octagon from side length.
Octagon Calculator
Calculator
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Formula: Area = 2(1 + sqrt(2)) x s^2
Worked example — Area: 754.44 sq in | Perimeter: 100 in | Span: 32.66 in
Formula
Area = 2(1 + sqrt(2)) x s^2
Where s is the side length of the regular octagon. The constant 2(1 + sqrt(2)) is approximately 4.8284. This formula derives from decomposing the octagon into triangles from the center or from a square minus corner triangles.
Worked Examples
Example 1: Stop Sign Dimensions
Problem:A standard US stop sign has sides of 12.5 inches. Calculate its area, perimeter, and the distance across the sign (long diagonal).
Solution:Side length s = 12.5 inches Area = 2(1 + sqrt(2)) x 12.5^2 = 2 x 2.4142 x 156.25 = 754.44 sq inches Perimeter = 8 x 12.5 = 100 inches Circumradius = 12.5 / (2 x sin(pi/8)) = 12.5 / 0.7654 = 16.332 inches Long diagonal = 2 x 16.332 = 32.664 inches Apothem = 12.5 x (1 + 1.4142) / 2 = 15.089 inches
Result:Area: 754.44 sq in | Perimeter: 100 in | Span: 32.66 in
Example 2: Octagonal Gazebo Foundation
Problem:An octagonal gazebo has sides of 1.5 meters. Calculate the floor area and the distance from center to each corner post.
Solution:Side length s = 1.5 m Area = 2(1 + sqrt(2)) x 1.5^2 = 4.8284 x 2.25 = 10.864 sq meters Circumradius = 1.5 / (2 x sin(pi/8)) = 1.5 / 0.7654 = 1.960 m Apothem = 1.5 x (1 + 1.4142) / 2 = 1.811 m Perimeter = 8 x 1.5 = 12.0 m
Result:Floor Area: 10.86 sq m | Center to corner: 1.96 m | Perimeter: 12.0 m
Frequently Asked Questions
What is a regular octagon and what are its properties?
A regular octagon is an eight-sided polygon with all sides equal in length and all interior angles equal. Each interior angle measures exactly 135 degrees, and the sum of all interior angles is 1,080 degrees. A regular octagon has 20 diagonals and exhibits 8 lines of symmetry. It has rotational symmetry of order 8, meaning it looks identical after rotation by 45 degrees. The octagon is one of the most common polygons in everyday life, appearing in stop signs, architectural designs, decorative tiles, and MMA fighting rings. Its near-circular shape makes it an efficient use of space while maintaining flat edges that are easier to construct than curves.
How do you calculate the area of a regular octagon?
The area of a regular octagon with side length s is calculated using the formula: Area = 2(1 + sqrt(2)) x s squared. This formula derives from dividing the octagon into eight identical isosceles triangles meeting at the center, or alternatively by starting with a square and accounting for the corner cuts. The constant 2(1 + sqrt(2)) equals approximately 4.8284. You can also calculate the area using the apothem: Area = (1/2) x perimeter x apothem = (1/2) x 8s x a, where a is the perpendicular distance from center to side midpoint. For a regular octagon with side length 10, the area would be 2(1 + 1.4142) x 100 = 482.84 square units. This formula only applies to regular octagons where all eight sides are equal.
What is the difference between apothem and circumradius?
The apothem and circumradius are two different measurements from the center of a regular polygon. The apothem is the perpendicular distance from the center to the midpoint of any side, essentially the shortest distance from center to edge. The circumradius is the distance from the center to any vertex, which is the radius of the circumscribed circle that passes through all vertices. For a regular octagon, the apothem equals s(1 + sqrt(2))/2 and the circumradius equals s/(2 sin(pi/8)). The circumradius is always larger than the apothem. These measurements are important in construction and design because the apothem determines the width between parallel sides, while the circumradius determines the overall span from corner to corner.
How are octagons used in architecture and construction?
Octagons appear extensively in architecture due to their aesthetic appeal and structural properties. The most famous octagonal structure is the Tower of the Winds in Athens (circa 50 BC), and octagonal floor plans appear in many baptisteries, domes, and towers throughout history. In modern construction, octagonal shapes are used for gazebos, hot tub surrounds, deck platforms, and window designs. The octagon is preferred over other polygons because it closely approximates a circle while using straight edges that are easier to build. In tiling, regular octagons combine with squares to create the truncated square tiling pattern, commonly seen in bathroom floors and decorative surfaces. Builders use octagon calculators to determine material requirements and precise measurements for cutting angles.
How many diagonals does an octagon have and what types exist?
A regular octagon has exactly 20 diagonals, calculated using the formula n(n-3)/2 where n equals 8. These diagonals come in three distinct types based on how many vertices they skip. Short diagonals connect vertices separated by one vertex (there are 8 of these), with length s(1 + sqrt(2)). Medium diagonals connect vertices separated by two vertices (there are 8 of these), with length s times sqrt(4 + 2sqrt(2)). Long diagonals connect opposite vertices directly through the center (there are 4 of these), with length equal to twice the circumradius. The long diagonals are the only ones that pass through the center point and they divide the octagon into congruent sections. Understanding diagonal lengths is important for structural bracing calculations and decorative pattern design.
How do you cut a regular octagon from a square, and how much gets removed?
A regular octagon of side s fits exactly inside a square of side s times (1 + sqrt(2)), which is 2.4142 s. To make it, cut a right triangle off each corner whose two equal legs measure s divided by sqrt(2), about 0.7071 s; measured along the square edge that is 0.2929 times the square side from each corner. The four corner triangles together have area s squared, and subtracting that from the square leaves 2(1 + sqrt(2)) s squared, exactly the octagon area formula. This is the standard shop method for octagonal posts and table tops.
What miter angle do you cut for an octagonal frame or deck rim?
Each interior angle of a regular octagon is 135 degrees, so two mating pieces must each be cut at half of that, 67.5 degrees, measured from the edge of the board. Most miter saws are marked from square rather than from the fence line, so the setting you dial in is 90 minus 67.5, that is 22.5 degrees. Eight joints at 22.5 degrees close the ring because the exterior angles must total 360. Small errors compound around eight cuts, so test the setting on scrap and check the closing gap before cutting stock.
How closely does a regular octagon approximate a circle?
Very closely by area, less so by outline. An octagon inscribed in a circle of radius R has area 2 sqrt(2) R squared, about 2.828 R squared, against pi R squared for the circle, so it captures just over 90 percent of the circle area. Its perimeter, 8s, reaches about 97.5 percent of the circumference. That is why octagons are used where a near-round shape is wanted but curved cutting is impractical, such as gazebo decks, planter surrounds, and column wraps, and why the flat-to-corner difference is still large enough to matter for clearances.
What turning angle do you use to draw an octagon, and what is the central angle?
Each exterior angle is 45 degrees, and exterior angles of any polygon sum to 360, so drawing an octagon means repeating the pair of instructions move forward s, turn 45 degrees, eight times. The central angle, formed at the centre by two radii to adjacent vertices, is also 360 divided by 8, or 45 degrees, which is why the octagon splits into eight congruent isosceles triangles. Those two 45 degree figures are the basis for both the plotting method and the area derivation from triangles.
How do you find the area if you only know the width across flats or the circumradius?
Width across flats is twice the apothem a, so a = W/2 and the area is 8 a squared times tan(22.5 degrees), which is 3.3137 a squared. Starting instead from the circumradius R, the distance from centre to a vertex, the area is 2 sqrt(2) R squared, or 2.8284 R squared. Both agree with the side-length formula: an octagon with side 10 has apothem 12.071 and circumradius 13.066, and all three routes return 482.84 square units. Machinists usually quote across flats, while layout work usually starts from the circumradius.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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