Circumscribed Figures

Definition

Circumscribed figures are geometric shapes that are drawn around another shape such that all the vertices of the inner shape touch the outer shape. In the context of circles, a circumscribed figure refers to a polygon whose vertices lie on the circumference of a circle, meaning it is inscribed within the circle. Understanding circumscribed figures involves recognizing their relationships with radius, diameter, and center points.

5 Must Know Facts For Your Next Test

1. A polygon can be circumscribed around a circle if it is possible to draw a single circle that touches each vertex of the polygon exactly once.
2. For any triangle, there exists a unique circumcircle that can be drawn, which means all triangles are circumscribed figures with respect to their circumcircle.
3. The center of the circumcircle for a triangle is known as the circumcenter, and it can be found by constructing perpendicular bisectors of the triangle's sides.
4. In a circumscribed polygon, each side must be tangent to the circle at exactly one point.
5. The radius of the circumcircle can be calculated using specific formulas depending on the type of polygon and its dimensions.

Review Questions

• How do circumscribed figures relate to inscribed angles and central angles in a circle?
  • Circumscribed figures are closely related to inscribed angles and central angles because they help illustrate how different angles function in relation to a circle. An inscribed angle is formed when two chords meet at a point on the circumference, while a central angle is created by two radii extending from the center to points on the circle. When studying circumscribed polygons, understanding these angles helps reveal important properties, like how inscribed angles measure half of their corresponding central angles.
• Discuss how to determine the circumcenter of a triangle and its significance in relation to circumscribed figures.
  • To determine the circumcenter of a triangle, you must construct the perpendicular bisectors of each side of the triangle. The point where these bisectors intersect is the circumcenter, which serves as the center for the triangle's circumcircle. The significance of this point lies in its ability to provide a unique circumcircle for any triangle, making it an essential concept when working with circumscribed figures.
• Evaluate how understanding circumscribed figures enhances your comprehension of geometric relationships and properties.
  • Understanding circumscribed figures enhances comprehension of geometric relationships by highlighting how polygons and circles interact. When you recognize that certain polygons can have unique circumcircles, it opens up discussions about angles, tangents, and distances from vertices to centers. This knowledge not only allows for solving geometric problems more effectively but also deepens insights into congruence, similarity, and symmetry within various geometric shapes.

Related terms

Inscribed Angle:

An angle formed by two chords in a circle which share an endpoint; the vertex of the angle is on the circle.

Central Angle: An angle whose vertex is at the center of a circle and whose sides intersect the circle.

Circumcircle:

The circle that passes through all the vertices of a polygon, also known as the circumscribed circle.