Decagon Calculator - 10-Side Polygon Properties

A decagon calculator is a single-input geometry tool that solves a regular 10-sided polygon for the side length, perimeter, area, apothem, circumradius, all four distinct diagonals, and the interior and exterior angles, all from one number you already know.

• • Geometry homework and exam checks: Confirm a worked regular decagon problem, verify a closed-form area or diagonal answer, or check a computed side length without re-deriving the formula.
• • Architecture and landscaping layouts: Plan a decagonal gazebo, pool, fountain, or garden bed by turning one side into the inside area, perimeter, apothem, and circumradius for cut lists and material estimates.
• • Engineering, crafts, and design work: Size a decagonal nut, bolt head, badge, coin, patio, or inlay against the inscribed and circumscribed circle radii and all four diagonal lengths for fit and tolerance work.
• • Quick property cross-checks: Recover the side length from a known area, perimeter, circumradius, apothem, or any diagonal in one pass instead of working backward from the closed-form formulas by hand.

Most users only know one decagon measurement when they start, and a tool that only accepts the side length forces a manual round trip back to a. This decagon calculator skips that step: enter the perimeter, area, apothem, circumradius, or any diagonal, and the side length is solved for internally before the rest of the result panel is filled in.

If you only need the area and a few side properties from the side length, the dedicated Decagon Area Calculator keeps that one-input, one-output workflow.

The decagon calculator converts the chosen input to meters (or square meters for area), solves for the side length a, evaluates the closed-form decagon formulas for every other property, and then re-expresses the linear outputs in the chosen length unit and the area in the chosen area unit.

• a (side length): Length of one of the ten equal sides; the single unknown the calculator solves for first.
• phi (golden ratio): (1 + sqrt(5)) / 2, about 1.6180339. The circumradius is a times phi, and d5 is a times 2 phi.
• n = 10: Number of sides. The closed-form area coefficient, the apothem ratio, and the four diagonal coefficients all derive from the regular-polygon identities with n fixed at 10.
• input mode: Which of side, perimeter, area, apothem, circumradius, or any diagonal you typed, so the tool knows which inverse coefficient to divide by to recover a.

When the input is an area, the calculator divides by the closed-form coefficient 7.6942088 and takes the square root. When the input is a diagonal, it divides by the matching coefficient to recover a.

According to Wolfram MathWorld, the closed-form decagon area is A = (5/2) a^2 sqrt(5 + 2 sqrt 5) and the circumradius is R = a (1 + sqrt 5) / 2 = phi a

According to Britannica, a decagon is a ten-sided polygon with equal sides and angles and area A = (5/2) a^2 cot(pi/10)

For any regular n-sided polygon, the general formula A = n a^2 / (4 tan(pi/n)) drives the Polygon Area Calculator, which takes the same single-input approach and lets you change the number of sides.

Four short ideas explain every number in the result panel and prevent the most common mix-ups when you work with a regular ten-sided polygon.

Knowing the four diagonal lengths matters when you are cutting stencils or sizing a window that fits inside a decagonal frame.

The same ten-triangle decomposition with n = 6 instead of 10 drives the Hexagon Calculator, so the apothem and interior angle pattern is the closest comparison point for a six-sided peer.

Use the decagon calculator in five short steps, switching length or area units or input mode at any time without re-entering the rest of the data.

1. 1 Type your known decagon measurement: Enter the value you already have into the Known Measurement field. The unit is set in the Length Unit dropdown below.
2. 2 Pick what the value represents: In the 'This Value Represents' dropdown, choose Side length, Perimeter, Area, Circumradius, Apothem, or any of the four diagonals (d2, d3, d4, d5).
3. 3 Choose the length and area units: Pick a length unit (mm, cm, m, in, ft, yd) and an area unit (cm^2, m^2, in^2, ft^2). The length unit applies to every linear output, the area unit to the area row.
4. 4 Read the result panel: The side, perimeter, area, apothem, circumradius, all four diagonals, and both angles appear in one view. The interior and exterior angles are always 144 deg and 36 deg.
5. 5 Reset or change inputs: Hit Reset to restore the defaults, or change any field and the rest of the panel updates without reloading. Switching input modes preserves the side length and re-labels the rest.

For an exam problem that gives the area of a decagon as 100 m^2 and asks for the side length, enter 100 in the Known Measurement field, set 'This Value Represents' to Area, leave the units on meters, and read the side length and all diagonals in the same panel.

A decagon-specific tool with all ten properties in one panel is faster and less error-prone than a generic polygon formula sheet, especially when the input is not the side length.

• • One input, every property: Type the side, perimeter, area, apothem, circumradius, or any of the four diagonals, and the rest of the panel fills in.
• • Length and area units built in: Switch between mm, cm, m, in, ft, and yd for the linear outputs and between cm^2, m^2, in^2, and ft^2 for the area.
• • Closed-form precision: The closed-form area coefficient 7.6942088 and the four diagonal coefficients come from the standard regular n-gon identities, so results match Wolfram MathWorld to four decimal places by default.
• • Useful for visual or numeric checks: The four distinct diagonal lengths (d2, d3, d4, d5) plus apothem and circumradius are the values you actually need to lay out a decagon by hand, so the result panel doubles as a cutting or scale-drawing reference.
• • Angle sanity check: Interior and exterior angles are fixed at 144 deg and 36 deg, so they read like a built-in unit test against the rest of the panel.

When the inscribed and circumscribed circles matter more than the polygon sides, the Circle Calculator handles the diameter and radius pair that the apothem and circumradius are tied to.

Three inputs and two structural properties determine what comes out of the result panel. Knowing which lever does what prevents the most common mix-ups.

• • The tool assumes a regular decagon only. An irregular ten-sided polygon needs a different approach.
• • Very small side lengths below 1e-6 m can lose precision in the result panel because the area scales with the square of the side and the diagonals are linear multiples.
• • Inputs are limited to non-negative values. The decagon as a closed polygon has no negative or zero side, so the form rejects them and leaves the result panel unchanged.

According to Wikipedia, a regular decagon has 10 sides, 10 interior angles of 144 deg, 10 exterior angles of 36 deg, and 35 diagonals of four distinct lengths

The same workflow with n = 12 lives in the Dodecagon Area Calculator, and the side-by-side dodecagon-decagon comparison is the cleanest way to see how the regular n-gon formulas scale with the number of sides.