An inscribed polygon is a polygon placed inside a circle so that every one of its vertices touches the circle’s edge. The circle that passes through all the vertices is called the circumscircle, and the polygon is said to be “inscribed” in it. This concept shows up throughout geometry, from basic triangle properties to methods for approximating pi.
How Inscribed Polygons Work
The key requirement is simple: every corner of the polygon must sit exactly on the circumference of the circle. A triangle with all three corners on a circle is inscribed. A square with all four corners on a circle is inscribed. A 100-sided polygon with every vertex on a circle is inscribed. If even one vertex falls inside or outside the circle, the polygon isn’t truly inscribed.
The circle itself has a center point that is equidistant from every vertex of the polygon. That distance is called the circumradius. For a triangle, you can find this center (called the circumcenter) by drawing the perpendicular bisector of each side. The three bisectors all cross at the same point, and that point is the center of the circumscribed circle.
Inscribed vs. Circumscribed Polygons
These two terms describe opposite arrangements of a polygon and a circle. An inscribed polygon sits inside the circle with its vertices on the circle’s edge. A circumscribed polygon sits outside the circle with each of its sides tangent to (just touching) the circle. In the circumscribed case, the circle is on the inside, and the polygon wraps around it.
Think of it this way: for an inscribed polygon, the circle is the boundary and the polygon fits within it. For a circumscribed polygon, the polygon is the boundary and the circle fits within it. The same circle can have a polygon inscribed inside it and a different polygon circumscribed around it at the same time.
Properties of Inscribed Triangles
Every triangle can be inscribed in a circle. This isn’t true for all polygons, but it’s always true for triangles. The circumcenter may fall inside the triangle (for acute triangles), on the hypotenuse (for right triangles), or outside the triangle (for obtuse triangles), but it always exists.
One of the most elegant results in geometry involves triangles inscribed in a semicircle. If one side of the triangle is the diameter of the circle, the angle opposite that side is always exactly 90°. This holds no matter where you place the third vertex on the semicircle. The result is known as Thales’ theorem, and it means any triangle inscribed in a semicircle is a right triangle.
Properties of Inscribed Quadrilaterals
A four-sided polygon inscribed in a circle is called a cyclic quadrilateral, and it has a property that regular quadrilaterals don’t necessarily share: opposite angles always add up to 180°. If one angle is 70°, the angle directly across from it is 110°. If one is 90°, the opposite is also 90°. This works for both pairs of opposite angles in the shape.
Cyclic quadrilaterals also follow a relationship called Ptolemy’s theorem, which connects the sides and diagonals. If the four sides are a, b, c, and d (in order), and the two diagonals are p and q, then ac + bd = pq. In plain terms, the sum of the products of opposite sides equals the product of the diagonals. This relationship only holds when the quadrilateral is inscribed in a circle.
Regular Inscribed Polygons and Formulas
A regular inscribed polygon has all sides equal and all angles equal, with every vertex on the circle. Regular inscribed polygons are especially useful because their geometry is predictable and follows clean formulas.
For a regular polygon with n sides inscribed in a circle of radius r, each side has a length of 2r sin(π/n). As you increase the number of sides, each individual side gets shorter, but the total perimeter grows and approaches the circumference of the circle. The full perimeter is 2nr sin(π/n).
This convergence toward the circle’s circumference is historically significant. Archimedes used inscribed and circumscribed polygons to estimate pi. By calculating the perimeters of 96-sided polygons both inside and outside a circle, he pinned pi between 3.1408 and 3.1429. The inscribed polygon’s perimeter is always slightly less than the circle’s circumference (since it cuts across the interior), while the circumscribed polygon’s perimeter is always slightly more (since it extends beyond the circle). Adding more sides narrows the gap.
Which Polygons Can Be Inscribed
Not every polygon can be inscribed in a circle. All triangles can, and all regular polygons (equilateral triangles, squares, regular pentagons, etc.) can. But irregular quadrilaterals and other shapes with uneven angles often cannot. A quadrilateral can only be inscribed if its opposite angles sum to 180°. If they don’t, there’s no single circle that passes through all four corners.
For polygons with five or more sides, the conditions get more complex. In general, you need a specific geometric relationship between the vertices for a circumscircle to exist. Regular polygons always satisfy these conditions because their symmetry guarantees that all vertices are the same distance from the center.