Decagon Area Calculator
Compute the area, perimeter, side length, circumradius, apothem and diagonals of a regular decagon — enter any one measurement.
Calculator
Results
Area A = – units²
| Measurement | Value |
|---|---|
| Side length s | – |
| Perimeter P = 10s | – |
| Area A | – |
| Circumradius R = s / (2 sin 18°) | – |
| Apothem / inradius a = s / (2 tan 18°) | – |
| Diagonal (2 edges) ≈ 1.9021130 × s | – |
| Diagonal (3 edges) ≈ 2.6180340 × s | – |
| Longest diagonal (through center) = 2R | – |
| Width across flats = 2a | – |
| Width across corners = 2R | – |
| Interior angle | 144° |
| Central / exterior angle | 36° |
| Number of diagonals | 35 |
Formulas
All angles: interior 144°, exterior 36°, central 36°. A decagon has n(n−3)/2 = 35 diagonals.
Worked examples
1. Side s = 1 m
A = 7.694208842938134 × 1² = 7.694208842938134 m², P = 10 m, R ≈ 1.618033988749895 m, apothem ≈ 1.538841768587627 m.
2. Side s = 5 cm
A = 7.694208842938134 × 25 = 192.35522107345335 cm², P = 50 cm, R ≈ 8.090169943749475 cm, apothem ≈ 7.694208842938135 cm.
3. Circumradius R = 10 m
s = 10 / 1.618033988749895 = 6.180339887498948 m; A ≈ 293.8926261462366 m², P ≈ 61.80339887498948 m, apothem ≈ 9.510565162951536 m.
4. Apothem a = 8 ft
s = 8 / 1.538841768587627 = 5.198715139726501 ft; A ≈ 207.94860558906004 ft², P ≈ 51.98715139726501 ft, R ≈ 8.411697793906137 ft.
Frequently asked questions
What is the formula for the area of a regular decagon?
For a regular decagon with side length s, the exact area is A = (5/2) × s² × √(5 + 2√5), approximately 7.694208842938134 × s². For example, a decagon with side 10 has area 769.4208842938134 square units.
How do you find the side length of a decagon from its area?
Rearrange the area formula: s = √(A / 7.694208842938134). A decagon of area 192.35522107345335 therefore has side s = 5.
What is the circumradius of a regular decagon with side s?
R = s / (2 × sin 18°) ≈ 1.618033988749895 × s — exactly the golden ratio φ times the side. A decagon of side 1 has circumradius ≈ 1.618033988749895.
What is the apothem (inradius) of a regular decagon?
a = s / (2 × tan 18°) ≈ 1.538841768587627 × s. Twice the apothem is the width across the flat sides.
What are the interior angles of a decagon?
Each interior angle is 144°, each exterior angle is 36°, and the central angle subtended by one side at the center is also 36°.
How many diagonals does a decagon have?
35 diagonals (10 × 7 / 2). In a regular decagon the 2-edge diagonal ≈ 1.9021130325903073 × s, the 3-edge diagonal ≈ 2.618033988749895 × s, and the longest (through the center) is 2R ≈ 3.23606797749979 × s.
Can I compute the area if I only know the apothem?
Yes: s = a / 1.538841768587627, then A ≈ 7.694208842938134 × s². An apothem of 8 ft gives side ≈ 5.198715139726501 ft and area ≈ 207.94860558906004 ft².
What is the perimeter of a regular decagon?
P = 10 × s. A decagon with side 6.180339887498948 has perimeter 61.80339887498948, corresponding to a circumradius of exactly 10.
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