Decagon Calculator - Area, Perimeter, Diagonals and Radii
Enter any single measurement of a regular decagon - side length, perimeter, area, circumcircle radius, incircle radius, or any of the four distinct diagonal lengths - and this calculator solves all remaining properties instantly. Results update as you type. Switch between metric and imperial units with one click.
What is a regular decagon?
A regular decagon is a flat, closed polygon with 10 sides of equal length and 10 interior angles of equal measure. Each interior angle is 144 degrees and each exterior angle is 36 degrees. Because of its high degree of symmetry, a regular decagon has 10 lines of reflective symmetry and rotational symmetry of order 10. It appears in architecture, design, coin shapes (the British 10 pence and 50 pence), and in various tiling patterns.
Decagon formulas: area, perimeter, circumradius and inradius
For a regular decagon with side length a, the key formulas are: Perimeter p = 10*a. Area A = (5/2) * a^2 * sqrt(5 + 2*sqrt(5)), approximately 7.6942 * a^2. Circumcircle radius R = a * (1 + sqrt(5)) / 2, which is the golden ratio times a (approximately 1.6180 * a). Incircle radius r = (a/2) * sqrt(5 + 2*sqrt(5)), approximately 1.5388 * a. The inradius is also called the apothem, and it is the perpendicular distance from the centre of the decagon to the midpoint of any side. The circumradius is the distance from the centre to any vertex.
The four distinct diagonals of a regular decagon
A regular decagon has 10*(10-3)/2 = 35 diagonals in total, but because of the high symmetry there are only four distinct lengths. Using the chord formula chord_k = a * sin(k*pi/10) / sin(pi/10), where k is the number of vertices skipped: the short diagonal d2 (k=2) is approximately 1.9021 * a; the medium diagonal d3 (k=3) is approximately 2.6180 * a, which equals twice the circumradius divided by the golden ratio; the long diagonal d4 (k=4) is approximately 3.0777 * a, equal to twice the inradius; and the longest diagonal d5 (k=5) is approximately 3.2361 * a, equal to 2*R and equal to 1/sin(18 degrees) * a. This longest diagonal is the diameter of the circumscribed circle and passes through the centre of the decagon. The ratios among the diagonal lengths are deeply connected to the golden ratio because the regular decagon and regular pentagon share the same circumscribed circle when their vertices are aligned.
How to use this calculator
Select which property you already know from the "Known property" dropdown. Enter its value in the field below. All other properties are computed and displayed immediately. Switch between metric (centimetres) and imperial (inches) units with the unit selector at the top. The "Show your work" panel below the results explains every calculation step with the actual numbers you entered. The bar chart to the right of the results compares all linear dimensions at a glance, and the line chart at the bottom shows how the area and perimeter scale as the side length changes.
Regular decagon: multipliers relative to side length a
| Property | Formula | Multiplier (x a or x a^2) |
|---|---|---|
| Perimeter | 10 * a | 10.000000 |
| Area | (5/2) * a^2 * sqrt(5+2*sqrt(5)) | 7.694209 (x a^2) |
| Circumradius R | a * (1+sqrt(5))/2 | 1.618034 |
| Inradius r | (a/2) * sqrt(5+2*sqrt(5)) | 1.538842 |
| Diagonal d2 | a * sin(36) / sin(18) | 1.902113 |
| Diagonal d3 | a * sin(54) / sin(18) | 2.618034 |
| Diagonal d4 | a * sin(72) / sin(18) | 3.077684 |
| Longest diagonal d5 | a * sin(90) / sin(18) = a/sin(18) | 3.236068 |
| Interior angle | (10-2)*180 / 10 | 144 degrees |
| Exterior angle | 360 / 10 | 36 degrees |
Multiply the side length by these constants to get each property. All values are exact or rounded to 6 significant figures.
Frequently asked questions
What is the area of a regular decagon?
The area of a regular decagon with side length a is A = (5/2) * a^2 * sqrt(5 + 2*sqrt(5)), which simplifies to approximately 7.6942 * a^2. For example, a decagon with a 5 cm side has an area of about 7.6942 * 25 = 192.4 cm^2.
How many diagonals does a decagon have?
A regular decagon has 35 diagonals in total, calculated as n*(n-3)/2 = 10*7/2 = 35. However, because of symmetry, there are only four distinct diagonal lengths, ranging from about 1.90 times the side length to about 3.24 times the side length.
What is the apothem of a decagon?
The apothem is the perpendicular distance from the centre of the decagon to the midpoint of any side. It equals the incircle radius r = (a/2) * sqrt(5 + 2*sqrt(5)) ≈ 1.5388 * a. You can also compute the area as A = (1/2) * perimeter * apothem.
What is the longest diagonal of a regular decagon?
The longest diagonal connects two vertices that are directly opposite each other (5 vertices apart) and passes through the centre of the decagon. Its length is a / sin(18 degrees) ≈ 3.2361 * a, which equals twice the circumcircle radius (2R). This diagonal is the diameter of the circumscribed circle.
How is the golden ratio related to the decagon?
The golden ratio phi = (1 + sqrt(5)) / 2 ≈ 1.6180 appears naturally in the decagon because the decagon and the regular pentagon share the same geometry. The circumcircle radius equals phi times the side length. The diagonal d3 equals 2 * phi * a. Several of the diagonal ratios are powers or products of the golden ratio.
Can I enter area instead of side length?
Yes. Select "Area (A)" from the "Known property" dropdown, enter the area value, and the calculator will back-solve the side length using a = sqrt(A / 7.6942), then derive all other properties from it. Every property listed in the dropdown can be used as the starting point.