Decagon Area Calculator - 10-Side Polygon Area

A decagon area calculator is a single-input tool that takes one side length of a regular 10-sided polygon and returns the inside area, perimeter, apothem, circumradius, and the interior and central angles, so you can stop deriving A = (5/2) s^2 sqrt(5 + 2 sqrt 5) by hand for craft, classroom, or project planning. Enter a side in any common linear unit, pick the area unit you want, and the tool applies the closed-form decagon formula and the ten equal sides to give you all six numbers in one pass.

• • Geometry homework and exams: Verify a worked regular decagon problem, confirm a closed-form solution, or check a unit conversion.
• • Crafts, tiles, and tabletop layouts: Estimate the decagon-shaped area of a patio, gazebo floor, custom tabletop, or inlay pattern when you only know the side length.
• • Architecture and landscaping: Plan a decagon-shaped pool, gazebo, fountain surround, or garden bed by turning the side length into the area the surface will cover.
• • Engineering and product design: Get the apothem and circumradius to size the inscribed and circumscribed circles of a decagonal nut, bolt head, badge, or coin before drafting.

If you want the same area, perimeter, apothem, and circumradius for any regular polygon other than a decagon, the polygon area calculator takes the same one-input approach and lets you change the number of sides.

The tool converts the side length to meters, evaluates the closed-form decagon area formula, derives the apothem, circumradius, perimeter, and angles from the same geometry, and then re-expresses the linear outputs in your chosen length unit and the area in the chosen area unit.

• s (side length): Length of one of the ten equal sides, entered in the chosen length unit (mm, cm, m, in, ft, yd).
• pi: The mathematical constant pi (approximately 3.14159265), used in the central angle and the tangent or sine of pi/10.
• sqrt(5): Square root of 5, approximately 2.2360679, which appears in the closed-form area, apothem, and circumradius formulas.

Each of the ten triangles formed by the center, two adjacent vertices, and one side has the same height (the apothem) and the same base (one side length). The area of the full decagon is 10 times the area of one such triangle, and that relationship gives both the area formula A = (1/2) * P * a and the link to the inscribed and circumscribed circles.

According to Wolfram MathWorld, a regular decagon has 10 equal sides and area A = (5/2) * s^2 * sqrt(5 + 2*sqrt(5)), which equals 7.6942088 * s^2.

According to Math Open Reference, the area of a regular decagon with side length s is A = (5/2) * s^2 * sqrt(5 + 2*sqrt(5)), the apothem is s / (2 * tan(pi/10)), and the circumradius is s / (2 * sin(pi/10)).

The same regular-polygon formula with n = 6 instead of n = 10 drives the hexagon calculator, so the worked example, apothem ratio, and interior angle pattern are the closest comparison point for a six-sided peer.

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.

As the number of sides grows past 10, the regular polygon approaches a circle, and the circle calculator takes the same radius input and applies the limiting pi r^2 formula to the apothem or circumradius result.

Use the decagon area calculator in five short steps, switching length or area units at any time without re-entering the side length.

1. 1 Enter the side length: Type the length of one side of the regular decagon into the Side Length field. The value must be a positive number; the unit is set in the dropdown below.
2. 2 Pick the length unit: Select millimeters, centimeters, meters, inches, feet, or yards so the tool can convert the side length to meters internally. Perimeter, apothem, and circumradius are reported in the same unit.
3. 3 Pick the area unit: Select square meters, square centimeters, square feet, or square inches for the area readout. The choice does not change the physical area, only the displayed number.
4. 4 Read the result panel: Watch the decagon area, perimeter, apothem, circumradius, interior angle, and central angle update in real time as you change the side length or any unit.
5. 5 Use the values in your project: Cross-check the apothem and circumradius against any drawing or template, and switch units to compare against a plan measured in feet, inches, centimeters, or meters.

If you are laying out a 1.2 m decagonal gazebo floor, type 1.2 into the side length field, leave the length unit on meters, and keep the area unit on square meters. The result panel returns about 11.08 m^2 of inside area, 12 m of perimeter, 1.8466 m of apothem, and 1.9416 m of circumradius, which is enough floor area for a small seating layout and matches the inside and outside diameters you would cut on a miter saw.

A dedicated decagon area calculator removes the algebra from six tightly related numbers and keeps the units consistent for both metric and imperial projects.

• • Six decagon values in one pass: Enter one side length and read off the area, perimeter, apothem, circumradius, interior angle, and central angle at the same time.
• • Closed-form and general formula cross-check: The tool evaluates both A = (5/2) * s^2 * sqrt(5 + 2*sqrt(5)) and A = 10 * s^2 / (4 * tan(pi/10)) internally, so the decagon result matches the regular polygon area formula in every test case.
• • Metric and imperial without re-entering: Switch the side length between mm, cm, m, in, ft, and yd, and switch the area output between m^2, cm^2, ft^2, and in^2, without re-typing the side length.
• • Apothem and circumradius included: The result panel reports both the inscribed and circumscribed circle radii, so you can size a coin, badge, or bolt head against the same drawing without a second tool.
• • Hand-off to other regular polygon tools: The result lines up with the hexagon and polygon area peers, since the same general regular polygon formula drives all three calculators.

For a quick sanity check on a 2D area from any shape, including the squares and rectangles that frame a decagon layout, the area calculator returns the area from the length and width in a single pass.

Three measurable factors control the precision of every decagon area result, and a few practical limits apply to any real ten-sided shape.

• • The decagon area calculator assumes a perfectly closed, planar, regular decagon. Real wood, tile, or metal shapes have rounded edges, joint gaps, material thickness, or minor measurement error, so include a small waste margin above the calculated area.
• • Angles, apothem, and circumradius assume a flat layout. If the decagon is curved onto a cone, dome, or other 3D surface, a different surface area tool is needed.
• • The calculator does not compute the side length backwards from a given area. To solve for s from A, divide the area by 7.6942088 and take the square root.

As published by Encyclopaedia Britannica, a regular decagon is a 10-sided polygon whose interior angle is 144 degrees and whose central angle is 36 degrees, so its area depends only on a single side length.

If you also need a side-by-side area for a rectangle that frames the decagon or a tile that fills the inside of one of its faces, the length width area rectangle calculator gives the matching l * w in square feet or square meters.