Incircle and Circumcircle of a Triangle

The incircle of a triangle touches all three sides, while the circumcircle passes through all three vertices. In brief: a 3–4–5 right triangle has inradius 1 and circumradius 2.5 in the same length unit. Its incenter comes from angle bisectors, whereas its circumcenter comes from perpendicular bisectors of sides.

These circles represent two different equal-distance conditions. The incircle centre is equally distant from the sides; the circumcircle centre is equally distant from the vertices. This guide explains their constructions, locations and radius formulas, with a complete example and checks that distinguish the incenter, circumcenter and other triangle centres.

What is the incircle of a triangle?

An incircle lies inside a nondegenerate triangle and is tangent to each of its three sides. Its centre is the incenter, and its radius is the inradius, commonly denoted r. The perpendicular distances from the incenter to all three side lines are equal to r.

Every nondegenerate triangle has exactly one incircle. The tangent points lie on the side segments, and the radii drawn to them are perpendicular to those sides. Equal tangent segments from a common vertex provide useful length relationships along the triangle’s boundary.

The incircle should not be confused with an excircle, which touches one side and extensions of the other two. A triangle has three excircles, but the ordinary incircle is the one inside the triangle. Unless an excircle is explicitly requested, use the internal angle bisectors and the interior centre.

How do you construct the incenter and incircle?

Construct the internal bisectors of two triangle angles. Their intersection is the incenter I; the third internal angle bisector passes through the same point. A point on an angle bisector is equally distant from the angle’s two side lines, which explains the equal-distance property.

From I, drop a perpendicular to one triangle side and mark its foot T. The distance IT is the inradius. Draw the circle centred at I with radius IT; it touches all three sides because I has the same perpendicular distance to each one.

Do not use the distance from I to a vertex as the radius. That distance reaches a corner, not a tangent point on a side. The perpendicular construction is essential, and the incenter always lies inside the triangle because the internal angle bisectors meet in the interior.

What is the circumcircle of a triangle?

A circumcircle passes through all three vertices of a nondegenerate triangle. Its centre is the circumcenter O, and its radius is the circumradius, usually denoted R. The distances OA, OB and OC from the centre to the three vertices are equal.

Construct the perpendicular bisectors of two sides; their intersection gives O. A point on a segment’s perpendicular bisector is equally distant from its endpoints, so the intersection is equally distant from all three vertices. The third perpendicular bisector passes through the same centre.

Draw the circle using the distance from O to any vertex. Unlike the incircle, the circumcircle does not touch the sides tangentially; the sides are chords of it. The circumcenter may lie inside, on the boundary of or outside the triangle depending on the triangle’s angles.

Where are the incenter and circumcenter located?

The incenter is always inside a nondegenerate triangle. The circumcenter lies inside an acute triangle, at the midpoint of the hypotenuse of a right triangle and outside an obtuse triangle. These locations follow from the intersections of the relevant bisectors, not from a general rule that every triangle centre must be internal.

For a right triangle, the circumcircle has the hypotenuse as a diameter. The angle opposite that diameter is a right angle by the inscribed-angle theorem, and the hypotenuse midpoint is equally distant from all three vertices. Therefore the circumradius is half the hypotenuse.

Other triangle centres have different constructions. Medians meet at the centroid, while altitudes meet at the orthocenter. They do not generally coincide with the incenter or circumcenter. In an equilateral triangle, symmetry makes these centres coincide, but that special case should not be assumed for an arbitrary triangle.

How do you calculate the inradius?

If a triangle has area Δ and semiperimeter s = (a + b + c)/2, its inradius is r = Δ/s. The equivalent area formula is Δ = rs. Side lengths a, b and c must use the same unit, and the semiperimeter is half the full perimeter.

To understand the formula, join the incenter to the vertices. This divides the triangle into three smaller triangles whose bases are a, b and c and whose perpendicular heights are all r. Their total area is ar/2 + br/2 + cr/2 = r(a + b + c)/2 = rs.

If all side lengths are known, Heron’s formula Δ = √[s(s − a)(s − b)(s − c)] can supply the area before calculating r. First check that the side lengths form a nondegenerate triangle. A zero or invalid area indicates that the incircle model does not apply as an ordinary positive-radius triangle circle.

How do you calculate the circumradius?

For a triangle with side lengths a, b, c and area Δ, the circumradius is R = abc/(4Δ). Another form is R = a/(2 sin A), where A is the angle opposite side a. These formulas connect the circumcircle with the extended sine rule.

For a right triangle with hypotenuse c, the simpler result R = c/2 applies. It agrees with the general formula because the area is ab/2 for the two legs, giving abc/[4(ab/2)] = c/2. Use the right-triangle structure when it is explicitly known.

Keep uppercase R for circumradius and lowercase r for inradius when both appear. The two circles have different centres and generally different sizes. Their formulas use different combinations of area and side lengths, so identifying the requested circle comes before substituting measurements.

How do you find both radii for a 3–4–5 triangle?

The side lengths 3, 4 and 5 form a right triangle because 3² + 4² = 5². Its area is Δ = 3 × 4/2 = 6 square units, and its semiperimeter is s = (3 + 4 + 5)/2 = 6 units. Therefore the inradius is r = 6/6 = 1 unit.

The circumradius is half the hypotenuse, R = 5/2 = 2.5 units. The general formula confirms this: abc/(4Δ) = 3 × 4 × 5/(4 × 6) = 60/24 = 2.5. Agreement between the two routes checks the arithmetic.

Place the right-angle vertex at (0, 0) and the other vertices at (3, 0) and (0, 4). The incenter is (1, 1), while the circumcenter is the hypotenuse midpoint (1.5, 2). Their different locations make clear that the two radius values describe distinct circles rather than alternate measurements of one circle.

1. Identify the triangle’s side lengths and angle type.
2. Calculate area and semiperimeter.
3. Use r = Δ/s for the inradius.
4. Use R = abc/(4Δ), or half the hypotenuse for a right triangle.
5. Check centre constructions and attach length units.

What mistakes should you avoid with triangle circles?

The background distinction is equal distance to lines versus equal distance to points. Angle bisectors produce the centre equidistant from sides, while perpendicular bisectors produce the centre equidistant from vertices. Remembering those geometric meanings is more reliable than memorising two similar centre names alone.

Do not substitute the full perimeter for the semiperimeter in r = Δ/s, and do not use an incenter-to-vertex distance as the inradius. For circumradius calculations, pair a side with its opposite angle if using the sine formula. A mismatched side-angle pair generally produces the wrong result.

Finally, do not reject an exterior circumcenter in an obtuse triangle or assume the circles share a centre except in special symmetric cases. Verify the defining conditions: the incircle must be tangent to all sides, and the circumcircle must pass through all vertices. Those checks identify the intended construction directly.