Circumscribed
Example
Problem: A triangle has vertices at the points where a circle of radius 5 cm touches them. Describe the relationship between the circle and the triangle.
Step 1: The circle passes through all three vertices of the triangle, meaning every vertex lies exactly on the circle.
Step 2: The circle is the smallest circle that can contain the entire triangle. No smaller circle passes through all three vertices.
Step 3: We say the circle is circumscribed about the triangle, and equivalently, the triangle is inscribed in the circle. The radius of this circle is called the circumradius.
R=5 cm
Answer: The circle with radius 5 cm is circumscribed about the triangle. It is the smallest circle enclosing the triangle, passing through all three vertices.
Why It Matters
Circumscribed figures appear throughout geometry when you need to find the smallest enclosing shape around a polygon. For instance, every triangle has a unique circumscribed circle (circumcircle), and its center and radius are used in trigonometry, construction problems, and proofs involving angle relationships.
Common Mistakes
Mistake: Confusing circumscribed with inscribed.
Correction: Circumscribed means the shape is drawn around the outside (think "circum" = around). Inscribed means drawn inside. A circumscribed circle goes around a polygon; an inscribed circle fits inside it.
Related Terms
- Circumcircle — The circle circumscribed about a polygon
- Circle — The most common circumscribed shape
- Inscribed — Opposite concept: drawn inside a figure
- Circumradius — Radius of the circumscribed circle
- Polygon — Shape commonly enclosed by a circumscribed circle