Circumscribed Polygon Calculator
Regular polygons around circles (polygon tangent to circle)
Enter Values
Key Formulas
- • Side: s = 2r tan(π/n)
- • Circumradius: R = r / cos(π/n)
- • Area = ½ × perimeter × r
Circumscribed Polygon
Understanding Circumscribed Polygons
A circumscribed polygon (or polygon circumscribed about a circle) is a regular polygon whose sides are all tangent to an inscribed circle. The circle fits perfectly inside, touching each side at exactly one point.
Key Relationships
| Property | Formula |
|---|---|
| Side Length | s = 2r tan(π/n) |
| Circumradius | R = r / cos(π/n) |
| Perimeter | P = n × s = 2nr tan(π/n) |
| Area | A = ½Pr = nr² tan(π/n) |
| R/r ratio | R/r = 1 / cos(π/n) = sec(π/n) |
Inscribed vs Circumscribed
Inscribed Polygon
- • Polygon inside circle
- • Vertices on circle
- • Circle is circumscribed
- • s = 2R sin(π/n)
Circumscribed Polygon
- • Polygon around circle
- • Sides tangent to circle
- • Circle is inscribed
- • s = 2r tan(π/n)
Special Cases
- Triangle (n=3): s = 2r√3, R = 2r
- Square (n=4): s = 2r, R = r√2
- Hexagon (n=6): s = 2r/√3, R = 2r/√3
Frequently Asked Questions
What does circumscribed mean?
Circumscribed means "drawn around." A circumscribed polygon has a circle inscribed inside it, with the circle tangent to all sides.
What is the inradius?
The inradius (r) is the radius of the inscribed circle - the circle that fits inside the polygon and touches all sides. It's the apothem of the polygon.
How do inradius and apothem relate?
For a circumscribed polygon, the inradius and apothem are the same - both are the perpendicular distance from the center to each side.
Why use tan instead of sin?
For inscribed polygons, vertices are on the circle (sin relates to the radius). For circumscribed, sides are tangent (tan relates to the tangent point).
How is the area calculated?
Area = ½ × perimeter × inradius. This works because the polygon divides into n triangles, each with base s and height r (the inradius).
What happens as sides increase?
As n approaches infinity, the circumscribed polygon approaches the circle. The area approaches πr², and the ratio R/r approaches 1.