Definition: The largest circle the will fit inside a polygon that touches every side

Try this
Adjust the regular polygon below by dragging any orange dot, or alter the number of sides. Note the behavior of the polygon's incircle.

The incircle of a regular polygon is the largest circle that will fit inside the polygon and touch each side in just one place (see figure above) and so each of the sides is a tangent to the incircle. If the number of sides is 3, this is an equilateral triangle and its incircle is exactly the same as the one described in Incircle of a Triangle.

The inradius of a regular polygon is exactly the same as its apothem. The formulas below are the same as for the apothem. Use the formula that uses the facts you are given to start.

where

*s* is the length of any side

*n* is the number of sides

*tan* is the tangent function calculated in **degrees** (see Trigonometry Overview)

where

*r* is the radius (circumradius)

*n* is the number of sides

*cos* is the cosine function calculated in **degrees** (see Trigonometry Overview)

It can happen in reverse however. You can start with a circle and draw an irregular polygon around it as in the figure on the right. This would be called a circumscribed polygon.

Some mathematicians consider the incircle to be the largest circle that will fit inside a polygon, without the requirement that it touches all the sides. Clearly, under this definition, it is always possible to draw such a circle.

- Polygon general definition
- Quadrilateral
- Regular polygon
- Irregular polygon
- Convex polygons
- Concave polygons
- Polygon diagonals
- Polygon triangles
- Apothem of a regular polygon
- Polygon center
- Radius of a regular polygon
- Incircle of a regular polygon
- Incenter of a regular polygon
- Circumcircle of a polygon
- Parallelogram inscribed in a quadrilateral

- Square
- Diagonals of a square
- Rectangle
- Diagonals of a rectangle
- Golden rectangle
- Parallelogram
- Rhombus
- Trapezoid
- Trapezoid median
- Trapezium
- Kite
- Inscribed (cyclic) quadrilateral

- Regular polygon area
- Irregular polygon area
- Rhombus area
- Kite area
- Rectangle area
- Area of a square
- Trapezoid area
- Parallelogram area

- Perimeter of a polygon (regular and irregular)
- Perimeter of a triangle
- Perimeter of a rectangle
- Perimeter of a square
- Perimeter of a parallelogram
- Perimeter of a rhombus
- Perimeter of a trapezoid
- Perimeter of a kite

- Exterior angles of a polygon
- Interior angles of a polygon
- Relationship of interior/exterior angles
- Polygon central angle

- Tetragon, 4 sides
- Pentagon, 5 sides
- Hexagon, 6 sides
- Heptagon, 7 sides
- Octagon, 8 sides
- Nonagon Enneagon, 9 sides
- Decagon, 10 sides
- Undecagon, 11 sides
- Dodecagon, 12 sides

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