An inscribed circle is the largest circle that fits entirely inside a polygon, touching every side exactly once. You’ll most often encounter it in the context of triangles, where every triangle has exactly one inscribed circle (also called an incircle). The point where the circle touches a side is called the point of tangency, and the circle’s radius is known as the inradius.
How It Works in a Triangle
Every triangle, no matter its shape, has an inscribed circle. This circle sits inside the triangle and just barely touches all three sides. The key to finding it is the incenter, which is the single point where all three angle bisectors of the triangle meet. An angle bisector is a line that splits an angle into two equal halves. When you draw all three of these lines, they always cross at one interior point, and that point becomes the center of the inscribed circle.
The distance from the incenter to any side of the triangle is the same in every direction, and that distance is the radius of the circle. This is what makes the incircle special: it’s perfectly tangent to each side, meaning it touches without crossing. Because the incenter sits at equal distances from all three sides, it’s the natural center for the largest circle you could squeeze inside.
Calculating the Inradius
The radius of an inscribed circle in a triangle has a clean formula: divide the triangle’s area by its semiperimeter. The semiperimeter is just half the sum of all three side lengths. So if your triangle has sides of 5, 7, and 10, the semiperimeter is (5 + 7 + 10) / 2 = 11. Find the area using any method you like (Heron’s formula works well here), then divide by 11 to get the inradius.
Right triangles get an even simpler shortcut. If the two legs are a and b and the hypotenuse is c, the inradius equals (a + b − c) / 2. For a classic 3-4-5 right triangle, that gives (3 + 4 − 5) / 2 = 1. The inscribed circle has a radius of exactly 1 unit.
How to Construct One With a Compass
Drawing an inscribed circle by hand requires a compass and a straightedge. The process has a few steps, with one common mistake to watch for:
- Bisect two angles. Pick any two angles of the triangle and construct their angle bisectors. You only need two because two lines are enough to find a single intersection point inside the triangle.
- Mark the incenter. The point where the two bisectors cross is the incenter, the future center of your circle.
- Find the radius. This is where people often go wrong. You can’t just eyeball the radius. Construct a perpendicular line from the incenter to any one side of the triangle. The distance from the incenter to the point where that perpendicular meets the side is your exact radius.
- Draw the circle. Set your compass to that radius, place the point on the incenter, and sweep the circle. It should touch all three sides.
Inscribed Circles Beyond Triangles
Not every polygon can have an inscribed circle. A polygon that does is called a tangential polygon, meaning all its sides are tangent to one interior circle. Every triangle qualifies, but for quadrilaterals, there’s a specific requirement known as Pitot’s theorem: the sums of opposite side lengths must be equal. If a quadrilateral has sides a, b, c, and d (in order), it can have an inscribed circle only if a + c = b + d. The converse is also true: any convex quadrilateral meeting that condition will have an inscribed circle.
Squares and rhombuses always satisfy this condition, so they always have inscribed circles. Rectangles that aren’t squares do not. Regular polygons (equilateral triangles, squares, regular pentagons, regular hexagons, and so on) always have inscribed circles because their perfect symmetry guarantees equal distance from the center to every side.
Inscribed vs. Circumscribed Circles
These two concepts are easy to mix up because they both involve a circle and a polygon, but they work in opposite directions. An inscribed circle (incircle) sits inside the polygon and touches each side. A circumscribed circle (circumcircle) wraps around the outside of the polygon and passes through each vertex. The incircle’s center is found using angle bisectors, while the circumcircle’s center is found using perpendicular bisectors of the sides. In a triangle, the circumscribed circle’s center can actually land outside the triangle (this happens in obtuse triangles), but the inscribed circle’s center is always inside.
The two radii are also quite different in size. For an equilateral triangle with side length s, the circumradius is exactly twice the inradius. For other triangles, the ratio varies, but the circumradius is always larger.
Where Inscribed Circles Show Up
Inscribed circles aren’t just a textbook concept. In CAD software and architectural drafting, designers regularly use inscribed and circumscribed circles to define polygon dimensions. When you specify a hexagonal bolt head or a hexagonal window, the inscribed circle determines the “flat-to-flat” measurement (the distance between parallel sides), while the circumscribed circle gives the “corner-to-corner” measurement. Knowing which one you’re working with changes the physical size of the part or feature. In mechanical engineering, pipe fittings and hex nuts are routinely specified by the diameter of their inscribed circle because that’s the wrench size that fits across the flats.