Understanding the difference between circumscribed and inscribed figures is very helpful in geometry, especially when we talk about circles and shapes with straight sides, like triangles and squares. Let’s break down what each term means and how they’re different!

Definitions:

• Circumscribed Figure: This happens when a shape is drawn around a circle, and all the corners (or vertices) of the shape touch the circle. For example, a triangle can have a circle inside it that fits perfectly against all its sides.

• Inscribed Figure: This is the opposite! An inscribed figure is when a circle fits inside a shape, touching all its sides. Imagine a circle that snugly fits inside a triangle or another shape.

Key Characteristics:

1. Vertices and Edges:

   • In a circumscribed figure, the circle is called the circumcircle. The corners of the shape touch the circle.
   • In an inscribed figure, the circle is called the incircle. The sides of the shape just touch the circle at one point each.

2. How to Draw Them:

   • To draw a circumscribed figure, you need to find the center point of the shape (called the circumcenter) and then draw a circle that goes through all the corners.
   • For an inscribed figure, you find the point inside the shape (called the incenter) and draw a circle that touches all the sides.

Why It Matters:

• Circumscribed figures are often used in problems about triangles when we want to find something called the circumradius, which is related to the circle around the triangle.
• Inscribed figures help us find the inradius, which is related to the circle that fits inside the triangle.

Simple Formulas:

• For a triangle with sides a, b, and c, you can find the radius of the circumcircle (let’s call it R) using this formula:

  $R = \frac{abc}{4A}$

  Here, A is the area of the triangle.

• To find the radius of the incircle (we’ll call it r), you can use this formula:

  $r = \frac{A}{s}$

  In this case, s is the semiperimeter, which is half the perimeter of the triangle.

Being able to picture these ideas helps a lot in understanding how circles and shapes relate to each other. You’ll see these concepts in many geometry problems!