Inscribed Polygon Calculator
Regular polygons inscribed in circles
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Key Formulas
- • Side: s = 2R sin(π/n)
- • Apothem: a = R cos(π/n)
- • Area = ½ × perimeter × apothem
Inscribed Polygon
Understanding Inscribed Polygons
An inscribed polygon is a regular polygon whose vertices all lie on a circle (the circumscribed circle). The circle's radius is called the circumradius.
Formulas for Inscribed Regular Polygons
| Property | Formula |
|---|---|
| Side Length | s = 2R sin(π/n) |
| Apothem | a = R cos(π/n) |
| Perimeter | P = n × s = 2nR sin(π/n) |
| Area | A = ½Pa = ½nR² sin(2π/n) |
| Central Angle | θ = 360°/n |
Special Cases
- Triangle (n=3): s = R√3, Area = (3√3/4)R²
- Square (n=4): s = R√2, Area = 2R²
- Hexagon (n=6): s = R, Area = (3√3/2)R²
Approaching a Circle
As the number of sides increases, the inscribed polygon approaches the circle. The area approaches πR² and the perimeter approaches 2πR.
Frequently Asked Questions
What is an inscribed polygon?
An inscribed polygon has all its vertices lying exactly on the circumscribed circle. It fits inside the circle with each corner touching the circle.
What is the difference between inscribed and circumscribed?
An inscribed polygon fits inside the circle (vertices on circle). A circumscribed polygon has the circle inside it (sides tangent to circle).
What is the apothem?
The apothem is the perpendicular distance from the center to a side. For an inscribed polygon, a = R cos(π/n).
Why is hexagon special?
For a regular hexagon inscribed in a circle, the side length equals the radius (s = R). This makes hexagons easy to construct with a compass.
How do I find area using the apothem?
Area = (1/2) × perimeter × apothem. This works because the polygon can be divided into n congruent triangles with height = apothem.
What's the central angle?
The central angle is 360°/n, the angle at the center subtended by one side. It determines how the polygon is divided into congruent triangles.