How this instrument works
A regular decagon has ten equal sides and ten equal interior angles, so a single side length s determines everything else. The perimeter is simply ten times that one side, P = 10s. The area comes from a technique that works for any regular polygon: split the shape into identical isosceles triangles by drawing lines from the center to every corner, find one triangle's area, and multiply by the number of sides — for a decagon, that gives A = 10s² ⁄ (4·tan(π⁄10)).
Each of those ten triangles has an apex angle of exactly 360° ⁄ 10 = 36° at the decagon's center, and the tangent of half that angle (π⁄10 radians, or 18°) is exactly what lets a single triangle's area be expressed purely in terms of the side length s, with no separate apothem (center-to-edge distance) measurement needed as a starting input.
This same split-into-triangles approach generalizes directly to any regular polygon: an n-sided regular polygon's area is always n·s² ⁄ (4·tan(π⁄n)), with only the number of sides and the tangent's angle changing — a decagon is simply the case where n = 10.
- Enter the decagon's side length into the Side length field.
- Read Perimeter: ten times the side length.
- Read Area: the sheet applies the isosceles-triangle-decomposition formula automatically.
Worked example — a decagon with side 4
A regular decagon has a side length of 4. Its perimeter is 10 × 4 = 40, and its area is 10 × 16 ⁄ (4·tan(π⁄10)) ≈ 123.11 — the perimeter reached with plain multiplication, the area through the triangle-decomposition formula, both from that single measurement.
A larger decagon with side 6 has perimeter 60 and area about 276.99 — the perimeter scaling in direct proportion to the side length (1.5× larger), while the area scales with the SQUARE of that side length (2.25× larger), the same quadratic-versus-linear growth pattern every regular polygon shares.
Questions
What is a regular decagon?
A ten-sided polygon with all sides and all interior angles equal. Because of that regularity, a single side length is enough to determine both its perimeter and its area completely.
How is a decagon's area calculated from just the side length?
By splitting the decagon into ten identical isosceles triangles meeting at its center, each with an apex angle of 36° (360° divided by 10). One triangle's area, expressed purely in terms of the side length via the tangent of half that apex angle, is then multiplied by ten.
Does this formula work for other regular polygons?
Yes — the general pattern is A = n·s² ⁄ (4·tan(π⁄n)) for any regular n-sided polygon. A decagon is simply the case n = 10; the identical structure applies to a hexagon (n=6), heptagon (n=7), or any other regular polygon.
Does doubling the side length double the area?
No — area scales with the SQUARE of the side length, so doubling the side quadruples the area. Perimeter, by contrast, scales in direct proportion, so doubling the side simply doubles it.
What if the side length is zero?
The decagon has collapsed to a single point — both the perimeter and the area are exactly zero, a limit the formulas handle cleanly with no special case needed.