Summary of Side, Radius, and Apothem of Inscribed and Circumscribed Polygons

Keywords

What characterizes a polygon as inscribed or circumscribed?
How do we determine the relationship between a polygon's side and the radius of the circle it is inscribed in?
What is the definition of apothem and how does it relate to the radius in regular polygons?
Which properties of right triangles are used to establish these relationships?

Definition of inscribed and circumscribed polygons.
The importance of the circle's center and its relationship with vertices and sides of polygons.
The perpendicularity of the apothem with the side of the inscribed polygon.
Use of trigonometric relationships in right triangles to calculate radius, side, and apothem.

Relationship between side and radius (for inscribed polygons): l = 2 * R * sin(π/n)
Apothem calculation (for regular inscribed polygons): a = R * cos(π/n)
Calculation of the circumscribed circle radius: given L as the polygon's side, R = L / (2 * sin(π/n))
Calculation of the inscribed circle radius: given a as the apothem, r = a * tan(π/n)
Polygon perimeter: P = n * L
Polygon area: A = (P * a) / 2

Regular Polygons: Flat, convex geometric figures with all sides and angles equal.
Inscribed: A polygon is inscribed in a circle if all its vertices touch the circle's circumference.
Circumscribed: A circle is circumscribed around a polygon if the circumference touches all sides of the polygon.
Circle Radius (R): Distance from the circle's center to any point on the circumference.
Apothem (a): Perpendicular line segment from the center of a regular polygon to the midpoint of one of its sides.
Polygon Side (L): Each line segment that delimits the polygon.
Central Point: Point equidistant from all polygon vertices; it is also the center of the inscribed or circumscribed circle.
Perpendicularity: Relationship of a 90º angle between two lines, such as the apothem and the polygon side.
Right Triangles: Triangles that have a right angle (90º).

Understanding inscribed and circumscribed polygons is essential for visualizing their geometric relationships with the circle.
The perpendicular relationship between the apothem and the polygon side sets the basis for more precise calculations in geometry.
Properties of right triangles, such as the use of the Pythagorean Theorem and trigonometric relationships, are fundamental to understanding the relationships between sides, radii, and apothems.
The central point is key to building geometric relationships in regular polygons.

When inscribing a polygon in a circle, the circle's radius is the line connecting the center to any polygon vertex.
The apothem, in regular polygons, will always be perpendicular to a side, symmetrically dividing it.
Trigonometric relationships in the triangle formed by the radius, apothem, and half of a polygon side allow for measurements to be calculated without initially knowing all dimensions.
The perimeter of a regular polygon can be calculated by multiplying the side length by the total number of sides (P = n * L).
The polygon's area can be found from the perimeter and apothem, using the formula A = (P * a) / 2.

Inscribed Equilateral Triangle: An equilateral triangle inscribed in a circle forms three isosceles triangles sharing a central point, allowing calculations based on basic trigonometric relationships.
- Example: If we know the radius R, we can calculate the side L of the triangle using the formula L = 2 * R * sin(π/3).
Circumscribed Square: A circumscribed square has each side touching the circumference, with the circle passing through the midpoints of the square's sides.
- Example: If we know the side L of the square, the radius R of the circumscribed circle can be found by the relation R = L / (2 * sin(π/4)).
Regular Inscribed Hexagon: Each side of the hexagon is at an equal distance from the center, facilitating area and perimeter calculations.
- Example: Knowing the radius R, we calculate the apothem a = R * cos(π/6) and then obtain the perimeter P = 6 * L to find the area A = (P * a) / 2.

Regular inscribed and circumscribed polygons have special relationships with the containing circle, with the apothem and radius playing crucial roles in the geometry of these shapes.
The side of a regular inscribed polygon is proportional to the circle's radius and can be calculated using trigonometric functions based on the polygon's number of sides.
The apothem of a regular polygon is always perpendicular to the side and acts as a link between internal geometric properties of the polygon and the circumscribing circle.
We use right triangles and trigonometric relationships to derive formulas that relate the radius, apothem, and side of polygons inscribed in circles.

The relationship between the side of a regular polygon and the radius of an inscribed circle is given by the formula l = 2 * R * sin(π/n).
The apothem is calculated as a = R * cos(π/n) and is key to finding the polygon's area.
Through these relationships, it is possible to calculate the perimeter and area of the polygon without the need to directly measure all sides or the apothem.
By understanding these relationships, a variety of practical geometry problems are solved, reinforcing the importance of trigonometry in the study of geometric figures.