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Circumscribable

Circumscribable

Describes a plane figure that has a circumcircle.

Worked Example

Problem: Determine whether a rectangle with vertices at (0, 0), (6, 0), (6, 4), and (0, 4) is circumscribable.
Step 1: Find the center of the potential circumscircle. For a rectangle, the center is at the midpoint of a diagonal.
Center=(0+62, 0+42)=(3, 2)\text{Center} = \left(\frac{0+6}{2},\, \frac{0+4}{2}\right) = (3,\, 2)
Step 2: Compute the distance from the center to each vertex. Check the vertex (0, 0):
r=(3−0)2+(2−0)2=9+4=13r = \sqrt{(3-0)^2 + (2-0)^2} = \sqrt{9+4} = \sqrt{13}
Step 3: Check another vertex, say (6, 4):
r=(3−6)2+(2−4)2=9+4=13r = \sqrt{(3-6)^2 + (2-4)^2} = \sqrt{9+4} = \sqrt{13}
Step 4: By symmetry, all four vertices are the same distance from (3, 2). A circle of radius √13 centered at (3, 2) passes through every vertex.
Answer: Yes, the rectangle is circumscribable. Its circumcircle has center (3, 2) and radius √13.

Why It Matters

Knowing whether a polygon is circumscribable tells you that a unique circumscircle exists, which is essential in constructions, proofs, and calculations involving cyclic polygons. For instance, the extended law of sines relates a triangle's sides and angles directly to its circumradius, but only because every triangle is circumscribable.

Common Mistakes

Mistake: Confusing circumscribable with inscribable. Students sometimes think circumscribable means a circle can be drawn inside the figure.
Correction: Circumscribable means a circle passes through all vertices (circumscircle, outside). A figure with an inscribed circle (incircle, inside) is called inscribable or "tangential."

Related Terms