What Does Circumscribe Mean?
To circumscribe is to construct one figure around another so that it touches the inner figure at as many key points as possible. The word points outward: the circumscribing shape is on the outside.
Two everyday cases matter most:
A circumscribed circle (or circumcircle) is drawn around a polygon so it passes through all the polygon's vertices.
A circumscribed polygon is drawn around a circle so that each side is tangent to the circle (touching it at one point).
The opposite word is inscribe, which means to draw a figure inside another. A circle inscribed in a polygon touches its sides from within; a circle circumscribed about a polygon surrounds its corners. Both live inside the wider study of circles and polygons.
For an Example: Draw a circle through all three corners of any triangle, and it fits every single time.
Every triangle has exactly one such circle, no exceptions. Push past three sides, though, and that guarantee quietly disappears, which is where circumscribing gets interesting.
What Is a Circumscribed Circle (Circumcircle)?
A circumscribed circle of a polygon is the unique circle passing through every vertex of that polygon. Its centre is the circumcenter, and its radius is the circumradius, written $R$.
For a triangle, the circumcenter is the point where the three perpendicular bisectors of the sides meet, and the full construction is on the circumcenter of triangle page. A polygon whose vertices all lie on one circle is called concyclic, and such a polygon is a cyclic polygon.
Can Every Polygon Be Circumscribed by a Circle?
No, and this is the key idea people miss. Every triangle has a circumcircle, and so does every regular polygon, every rectangle, every isosceles trapezoid, and every right kite. But a general quadrilateral or irregular polygon usually does not have one, because its four or more vertices need not all sit on a single circle.
A polygon that does have a circumcircle is called cyclic; one that does not, simply is not.
How Do You Find the Circumradius?
The circumradius depends on the polygon. Two of the most useful formulas:
For any triangle with sides $a$, $b$, $c$ and area $K$:
$$R = \frac{abc}{4K}$$
For a right triangle, the circumcenter sits at the midpoint of the hypotenuse, so:
$$R = \frac{\text{hypotenuse}}{2}$$
For a regular polygon with $n$ sides of length $s$:
$$R = \frac{s}{2 \sin\left(\dfrac{180^\circ}{n}\right)}$$
Variable glossary: $a, b, c$ are triangle side lengths, $K$ is the triangle's area, $s$ is a regular polygon's side length, and $n$ is the number of sides. The radius idea is the same as for any circle; here it just reaches the polygon's vertices.
How Do You Construct a Circumscribed Circle?
Drawing a circumcircle around a triangle needs only a compass and a straightedge, because the circumcenter is where the perpendicular bisectors meet:
Draw the triangle and pick any two of its sides to work with.
Construct the perpendicular bisector of the first side. Open the compass to more than half the side's length, draw arcs from each endpoint above and below the side, and join the two arc crossings with a straight line.
Construct the perpendicular bisector of the second side the same way.
Mark the circumcenter. The two bisectors cross at a single point $O$ - this is the circumcenter. (The third side's bisector passes through $O$ too, so it makes a useful check.)
Draw the circle. Set the compass width to the distance from $O$ to any one vertex, then sweep a full circle. It passes through all three vertices at once - that is the circumscribed circle, and the compass width is the circumradius $R$.
Note: The perpendicular bisectors of two sides meet at the circumcenter $O$; a circle centred at $O$ through any vertex passes through all three.
Examples of Circumscribe
Example 1
Find the circumradius of a right triangle with legs $3$ and $4$ and hypotenuse $5$.
For a right triangle, the circumradius is half the hypotenuse:
$$R = \frac{5}{2} = 2.5 \text{ units}$$
Example 2
Find the circumradius of a triangle with sides $a = 6$, $b = 8$, $c = 10$.
Wrong attempt first. A student confuses circumradius with the inradius and uses the inradius formula $r = \frac{K}{s}$, where $s$ is the semi-perimeter.
Semi-perimeter $= \frac{6 + 8 + 10}{2} = 12$; area $K = \frac{1}{2}(6)(8) = 24$ (it is a right triangle). The student writes $r = \frac{24}{12} = 2$ and calls it the circumradius.
That is the inscribed circle's radius, not the circumscribed one. The context word "circumscribe" points outward, to the vertices, not inward to the sides.
Correct method. Use the circumradius formula:
$$R = \frac{abc}{4K} = \frac{6 \times 8 \times 10}{4 \times 24} = \frac{480}{96} = 5 \text{ units}$$
And since it is a right triangle, half the hypotenuse $= \frac{10}{2} = 5$ confirms it.
Example 3
Does a rectangle have a circumscribed circle? Find $R$ for a $6 \times 8$ rectangle.
Yes. A rectangle is cyclic, and its circumcircle's diameter is the diagonal.
Diagonal $= \sqrt{6^2 + 8^2} = \sqrt{100} = 10$, so:
$$R = \frac{10}{2} = 5 \text{ units}$$
Example 4
Find the circumradius of a square with side $s = 4$.
A square is a regular $4$-gon; its circumradius reaches from centre to corner, half the diagonal:
$$R = \frac{s\sqrt{2}}{2} = \frac{4\sqrt{2}}{2} = 2\sqrt{2} \approx 2.83 \text{ units}$$
Example 5
Find the circumradius of an equilateral triangle with side $a = 6$.
For an equilateral triangle, $R = \frac{a}{\sqrt{3}}$:
$$R = \frac{6}{\sqrt{3}} = 2\sqrt{3} \approx 3.46 \text{ units}$$
Example 6
A regular hexagon has side $s = 5$. Find the radius of its circumscribed circle.
For a regular hexagon ($n = 6$), $\sin\left(\frac{180^\circ}{6}\right) = \sin 30^\circ = \frac{1}{2}$:
$$R = \frac{5}{2 \times \frac{1}{2}} = 5 \text{ units}$$
A regular hexagon's circumradius equals its side length, a tidy special case.
Why Does Circumscribing Matter?
Circumscribing matters because the circumcenter is the one point equidistant from every vertex, which turns an abstract idea into a placement tool.
Facility placement. A tower serving three towns should sit at their circumcenter, the point equally far from all three, so coverage is balanced.
Engineering and design. Bounding circles (the smallest circle enclosing a shape's corners) are used in collision detection, machine vision, and part fitting.
Classical geometry. The result that any triangle is cyclic underpins the inscribed angle theorem and much of circle geometry, studied since Euclid's Elements around 300 BCE.
The deeper reason a triangle always has a circumcircle is that its three perpendicular bisectors always meet at one point, which is not guaranteed for four or more sides.
What Are the Most Common Mistakes With Circumscribe?
Mistake 1: Swapping circumscribed and inscribed
Where it slips in: any problem that names both an inner and outer circle.
Don't do this: treat the inscribed circle (inside, touching the sides) as the circumscribed one (outside, through the vertices).
The correct way: read the word direction. "Circum-" points outward to the vertices; "in-" points inward to the sides. The first instinct is to grab whichever circle is drawn most prominently, which is exactly the confusion the two words are built to prevent.
Mistake 2: Assuming every polygon has a circumcircle
Where it slips in: irregular quadrilaterals and general polygons.
Don't do this: assume you can always draw one circle through all the vertices.
The correct way: only cyclic polygons (all triangles, regular polygons, rectangles, isosceles trapezoids, right kites) are guaranteed one. A general four-sided shape usually is not cyclic.
Mistake 3: Using the inradius formula for the circumradius
Where it slips in: triangle problems, as in Example 2.
Don't do this: compute $\frac{K}{s}$ (the inradius) when the question asks for the circumradius.
The correct way: use $R = \frac{abc}{4K}$ for the circumscribed circle.
Conclusion
To circumscribe is to draw one figure around another so it touches it.
A circumscribed circle (circumcircle) passes through all a polygon's vertices; its centre is the circumcenter, its radius the circumradius $R$.
Every triangle and every regular polygon has a circumcircle, but a general polygon may not.
For a triangle, $R = \frac{abc}{4K}$; for a right triangle, $R$ is half the hypotenuse.
The most common mistake is confusing circumscribed with inscribed figures.
Practice These to Solidify Your Understanding
Try three: find the circumradius of a right triangle with hypotenuse $13$; find $R$ for an equilateral triangle of side $9$; and decide whether a $5 \times 12$ rectangle has a circumcircle, giving $R$ if it does. If you mix up the two circles, reread Example 2. To explore circle geometry with a teacher, try Bhanzu's geometry tutor, a high school math tutor, or math classes online. Want a live trainer to work through more constructions? Book a free demo class.
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