If for a polygon it is possible to construct an inscribed and circumscribed circle, then the area of this polygon is smaller than the circumscribed circumference, but larger than the area of the inscribed circle.
For some polygons, formulas are known for finding the radius of the inscribed and circumscribed circles.
Inscribed in a polygon is a circle touching all sides of a polygon. For a triangle, formula Of the radius Inscribed circle: r = ((p-a) (p-b) (p-c) / p) ^ 1/2, where p is the semiperimeter-a, b, c are the sides of the triangle. For a regular triangle, the formula is simplified: r = a / (2 * 3 ^ 1/2), and a is the side of the triangle.
The polygon described around thisA circle on which all the vertices of the polygon lie. For a triangle, the radius of the circumscribed circle is found by the formula: R = abc / (4 (p (p-a) (p-b) (p-c)) ^ 1/2), where p is the semiperimeter-a, b, c are the sides of the triangle. For a regular triangle the formula is simpler: R = a / 3 ^ 1/2.
For polygons it is not always possible to find outThe ratio of the radii of inscribed and circumscribed circumferences and the lengths of its sides. It is often limited to constructing such circles around a polygon, and then a physical measurement Of the radius Circles with the help of measuring instruments or vector space.
To construct the circumscribed circle of a convexPolygon construct bisectors of its two angles, at the intersection of which lies the center of the circumscribed circle. The radius is the distance from the point of intersection of the bisectors to the vertex of any corner of the polygon. The center of the inscribed circle lies at the intersection of perpendiculars built inward from the polygon from the centers of the sides (these perpendiculars are called the middle ones). It is enough to build two such perpendiculars. The radius of the inscribed circle is equal to the distance from the point of intersection of the median perpendiculars to the side of the polygon.