Solve a regular octagon from side length, area, perimeter, apothem, circumradius, diagonals, or bounding square width with area, perimeter, radius, fit.
Last updated
Regular octagon calculator Solve a regular octagon from side length, area, perimeter, apothem, circumradius,
any of the three diagonal lengths, or the square width used to construct it.
Known measurement
Quick examples
How this worksheet helps
Use this when one edge length of the regular octagon is known. The calculator converts that value back to side length, then derives area, perimeter, apothem, circumradius, diagonal lengths, and construction checks.
The fit rows show how the octagon compares with its bounding square and inner or outer circles, which is often more useful for layout than a formula alone.
Result
Side = 5
Solved from side length 5. Every regular-octagon measurement reduces back to side length, which is why one trusted value can solve the full shape.
Area
120.71
Perimeter
40
Across flats
12.07
Across points
13.07
Apothem
6.04
Circumradius
6.53
Short diagonal
9.24
Medium diagonal
12.07
Fit and construction checks
These rows connect the octagon to the square-and-corner-cut construction, plus the inner and outer circles used for clearance planning.
Comparison
Amount
Why it matters
Bounding square area
145.71
A regular octagon can be made by cutting four matching 45-degree corners from this square.
Total corner cut area
25
The four removed corner triangles together equal one side-length square.
Circumscribed circle area
134.08
This is the smallest circle that contains the octagon by passing through all eight vertices.
Octagon vs bounding square
82.84%
How much of the starting square remains after the four equal corner cuts.
Octagon vs outer circle
90.03%
Shows how closely the regular octagon approximates its circumscribed circle.
Inner circle vs octagon
94.81%
Useful when a circular insert must sit completely inside the octagon.
Scale checkpoints
This comparison makes the square-law growth visible before you resize an octagonal tile, sign, opening, or layout.
Scenario
Side
Area
Perimeter
Across flats
Across points
Meaning
Half scale
2.5
30.18
20
6.04
6.53
Linear dimensions halve, while area falls to one quarter.
Entered size
5
120.71
40
12.07
13.07
This is the solved regular octagon from your chosen starting measurement.
Double scale
10
482.84
80
24.14
26.13
Linear dimensions double, while area becomes four times as large.
Useful relationship A regular octagon has 20 diagonals, three diagonal lengths, 135-degree
interior angles, and 45-degree exterior and central angles. The medium diagonal equals
twice the apothem and the long diagonal equals twice the circumradius.
Regular octagon calculator from side, area, perimeter, apothem, radius, or diagonals
Use this octagon calculator to solve a regular octagon from side length, area, perimeter, apothem, circumradius, short diagonal, medium diagonal, long diagonal, or the bounding square width used in the classic corner-cut construction. It then turns that one known value into area, perimeter, all three diagonal lengths, inradius, circumradius, fit comparisons, and scale checkpoints for geometry, tiling, drafting, and layout work.
Why one regular-octagon measurement solves the whole shape
A regular octagon has eight equal sides and eight equal angles, so its geometry is locked once one compatible measurement is known. The most common starting value is side length, but real problems often start from area, perimeter, apothem, circumradius, a diagonal, or the square that the octagon was cut from. A stronger regular octagon calculator should support those practical entry points instead of forcing every user to convert to side length by hand.
This calculator follows the same reliable pattern for every solve mode: convert the known measurement back to side length, then derive area, perimeter, the three diagonal types, inradius, circumradius, and fit comparisons from that solved side. That workflow keeps the calculation consistent whether you are checking a worksheet, sizing an octagonal tile, laying out a sign, or comparing an octagon with a square or circle.
A = 2(1 + sqrt(2))s^2
Area of a regular octagon from side length s.
P = 8s
Perimeter from side length. This is the specific relationship the calculator applies when building the result.
r = (1 + sqrt(2))s / 2, R = sqrt(4 + 2sqrt(2))s / 2
Inradius or apothem r and circumradius R.
Octagon area, perimeter, apothem, and radii
The octagon area calculator uses A = 2(1 + sqrt(2))s^2, so the area is about 4.8284 times the square of the side length. The perimeter is simpler: multiply the side length by 8. These two headline results answer many classroom and quick-check questions, but they do not fully describe how the shape fits into a drawing or physical layout.
The apothem is the centre-to-side distance and is also the inradius of the largest circle that can fit inside the regular octagon. The circumradius is the centre-to-vertex distance of the smallest circle that contains all eight vertices. The calculator shows both because the apothem is useful for flat-to-flat or inscribed-circle work, while the circumradius is useful when the octagon is drawn from, or fitted inside, a circle.
Short, medium, and long octagon diagonals
A regular octagon has 20 diagonals, but they fall into three lengths. The short diagonal connects near non-adjacent vertices. The medium diagonal spans farther across the shape and equals the distance across opposite sides, so it is also twice the apothem. The long diagonal connects opposite vertices through the centre and is twice the circumradius.
This distinction matters because search results and worksheets often use similar words for different spans. If a drawing gives an across-flats measurement, use the medium diagonal or apothem relationship. If it gives the widest point-to-point dimension, use the long diagonal or circumradius relationship. If it gives the nearest internal diagonal in a tile pattern, use the short diagonal mode.
d_short = s sqrt(2 + sqrt(2))
The nearest non-adjacent vertex span. This is the specific relationship the calculator applies when building the result.
d_medium = s(1 + sqrt(2)) = 2r
The medium diagonal and opposite-side span.
d_long = s sqrt(4 + 2sqrt(2)) = 2R
The opposite-vertex span through the centre.
Worked example from side length
Suppose the side length is 5 units. The perimeter is 8 × 5 = 40 units. The area is 2(1 + sqrt(2)) × 5^2, which is about 120.71 square units. The apothem is about 6.04, the circumradius is about 6.53, the short diagonal is about 9.24, the medium diagonal is about 12.07, and the long diagonal is about 13.07.
Those values match the common examples shown by many octagon area calculator pages, but this page adds the context needed to use them. It also shows the across-flats span, across-points span, bounding square area, corner-cut area, inner-circle comparison, outer-circle comparison, and scale rows so the result can guide a real layout decision instead of stopping at a bare formula.
Side length 5 gives area about 120.71 square units
Perimeter becomes 40 units
Medium diagonal and across-flats span become about 12.07 units
Long diagonal and across-points span become about 13.07 units
Solving backward from area, perimeter, apothem, radius, or a diagonal
If you know the area, the calculator rearranges A = 2(1 + sqrt(2))s^2 to solve side length first. If you know perimeter, it divides by 8. If you know apothem, it uses s = 2r / (1 + sqrt(2)). If you know circumradius, it uses the long-diagonal relationship because the circumradius is half of that longest diagonal.
The same idea applies to diagonal inputs. Divide the short diagonal by sqrt(2 + sqrt(2)), divide the medium diagonal by 1 + sqrt(2), or divide the long diagonal by sqrt(4 + 2sqrt(2)). Supporting those reverse workflows makes the page useful for octagon perimeter calculator, octagon diagonal calculator, and regular octagon calculator searches without splitting the intent across several weaker pages.
Bounding square and corner-cut construction
A regular octagon can be built by starting with a square and cutting four equal 45-degree triangles from the corners. If the final octagon side is s, each corner-cut leg is s / sqrt(2), and the starting square width is s(1 + sqrt(2)). The total removed corner area equals s^2.
That construction is useful for drawing, woodworking, tiling, paper templates, and explaining why the area formula is compact. It also gives a quick fit check: the regular octagon fills about 82.84% of its bounding square. The calculator includes this comparison because it is hard to see from the side-length formula alone.
Circle fit and octagon efficiency
The inradius defines the largest circle that can sit entirely inside the octagon. The circumradius defines the smallest circle that can enclose the whole octagon. These two circles are useful when an octagonal shape needs clearance inside a round boundary or when a circular insert must fit inside an octagonal frame.
A regular octagon is also a close approximation to a circle for many layout purposes. Its area is roughly 90% of the circumscribed circle area, which is why octagonal layouts often feel more space-efficient than squares while remaining easier to construct than curves. The calculator reports that ratio directly so users do not need to run a separate circle-area calculation.
Angles, scaling, and common mistakes
Every regular octagon has 135-degree interior angles and 45-degree exterior angles. Linear measurements such as side length, perimeter, apothem, diagonals, and radii scale directly. Area scales with the square of the side length, so doubling the side length doubles the perimeter and diagonals but quadruples the area.
The most common mistakes are mixing up apothem with circumradius, treating the medium diagonal as the longest diagonal, and applying regular-octagon formulas to an irregular eight-sided polygon. This calculator assumes an ideal regular octagon only. If the sides or angles differ, one measurement is not enough to recover the full shape.
What this calculator does not cover
This tool handles plane-geometry regular octagons only. It does not solve irregular octagons, rounded-corner shapes, three-dimensional octagonal solids, material thickness, fabrication tolerances, or legal sign-design specifications.
Use the results as a geometry and planning worksheet. For construction, road-sign, product-design, or manufacturing work, confirm the actual drawing standard, unit convention, tolerance, and measurement definition before relying on a rounded calculator output.
Frequently asked questions
How do you calculate the area of a regular octagon?
Use A = 2(1 + sqrt(2))s^2 when the side length s is known. If the side length is not known, solve it first from the area, perimeter, apothem, circumradius, diagonal, or bounding square width, then apply the same area formula.
What is the perimeter formula for an octagon?
For a regular octagon, P = 8s because all eight sides are equal. For an irregular octagon, you must add all eight side lengths individually; one side length is not enough.
What is the difference between apothem and circumradius?
The apothem is the distance from the centre to a side and equals the inradius. The circumradius is the distance from the centre to a vertex. In a regular octagon, the circumradius is slightly larger than the apothem.
How many diagonals does an octagon have?
An octagon has 20 diagonals. In a regular octagon those diagonals fall into three lengths: short, medium, and long.
Which octagon diagonal is the longest?
The long diagonal is the vertex-to-opposite-vertex line through the centre. It is equal to twice the circumradius and is the same as the across-points span.
Is the medium diagonal the same as the apothem?
No. The medium diagonal is twice the apothem. It spans across opposite sides of the regular octagon, while the apothem is only the centre-to-side distance.
Can this calculator solve an octagon from area alone?
Yes. It rearranges the area formula to get side length with s = sqrt(A / (2(1 + sqrt(2)))), then calculates the perimeter, radii, diagonals, and fit comparisons from that side length.
Can this calculator solve from a diagonal length?
Yes. Choose short diagonal, medium diagonal, or long diagonal based on the measurement you have. The calculator divides by the correct diagonal factor to recover side length before solving the rest of the octagon.
How do you draw a regular octagon from a square?
Start with a square, then cut equal 45-degree triangles from all four corners. For a final side length s, each corner cut has leg length s / sqrt(2), and the starting square width is s(1 + sqrt(2)).
Why is a stop sign an octagon?
Traffic-control standards use the octagon as the distinctive stop-sign shape so the sign can be recognized by shape as well as colour and wording. This calculator can check octagon geometry, but it is not a sign-design compliance tool.
If I double the side length, what happens to the area?
The perimeter, apothem, circumradius, and diagonals double, but the area becomes four times as large. Area scales with the square of the side length.
Will this calculator work for an irregular octagon?
No. Regular-octagon formulas require all sides and all angles to be equal. An irregular octagon needs more measurements and a different geometry method.
