Inscribed and Circumscribed Polygons
When a polygon sits inside a circle with all its vertices touching the circle, it's called an inscribed polygon. The area of such a polygon can be found using the formula: , where r is the radius of the circle and n is the number of sides.
A circumscribed polygon wraps around a circle with all its sides touching the circle at exactly one point. For these polygons, the area formula is: , where r is the radius of the inscribed circle.
For triangles specifically, you can find the radius of a circumscribed circle using where s = . Similarly, the radius of an inscribed circle in a triangle is .
Remember This! The sides of a regular polygon inscribed in a circle can be calculated using - this is super helpful when constructing or analyzing geometric shapes.
Circle measurements are also important to know: the circumference is or , and the area is or . The center of an inscribed circle is equidistant from all sides of the circumscribed polygon, while the apothem of the circumscribed polygon equals the radius of the inscribed circle.
