Calculadora de círculo inscrito

The Inscribed Circle Calculator finds the radius of the largest circle that fits inside a triangle and touches all three sides. This circle is called the incircle, and its radius is the inradius, denoted r. Its center, the incenter, is one of the four classical "triangle centers" (along with the centroid, circumcenter, and orthocenter). This tutorial explains where the formula r = Area / s comes from, walks through a worked example, and contrasts the incircle with the circumscribed circle.

What the incircle is

The incircle is the unique circle that is:

• Inscribed inside the triangle (entirely within its boundary)
• Tangent to all three sides (touches each side at exactly one point)
• The largest such circle (no bigger circle can fit while touching all three sides)

The center of the incircle, the incenter, is the point where the three angle bisectors of the triangle meet. Every triangle has exactly one incircle — and its center is equidistant from each of the three sides. That equal distance is the inradius r.

The formula r = Area / s

The inradius is computed from the triangle's area and semi-perimeter:

r = Area / s, where s = (a + b + c) / 2

This is one of the cleanest formulas in plane geometry. Here is the geometric reason it works.

Connect the incenter I to each of the three vertices. This divides the triangle into three smaller triangles, each with one side of the original as base and the incenter as apex. The height of each small triangle (the perpendicular distance from I to the base) is exactly the inradius r — because I is equidistant from all three sides by construction.

The area of each small triangle is (1/2) × base × r:

• Small triangle on side a: (1/2)(a)(r)
• Small triangle on side b: (1/2)(b)(r)
• Small triangle on side c: (1/2)(c)(r)

Sum of the three areas = original triangle's area:

Area = (1/2)(a + b + c)(r) = (s)(r)

Solving for r: r = Area / s. QED.

How the calculator computes Area

The calculator uses Heron's formula to find the triangle's area from the three side lengths:

Area = √(s(s − a)(s − b)(s − c))

where s is the same semi-perimeter as above. Heron's formula needs only the three sides — no angles required.

Worked example

Inputs: a = 3, b = 4, c = 5 (a 3-4-5 right triangle).

1. Semi-perimeter: s = (3 + 4 + 5) / 2 = 6.
2. Area (Heron): √(6 × 3 × 2 × 1) = √36 = 6. (The 3-4-5 triangle's area is 6, since it is a right triangle with legs 3 and 4 → area = (1/2)(3)(4) = 6.)
3. Inradius: r = Area / s = 6 / 6 = 1.

The 3-4-5 triangle has an inradius of exactly 1. This is a "nice" example because all numbers come out clean — a useful sanity check that the formula and the calculator agree.

Incircle vs circumcircle

The incircle (this calculator) sits inside the triangle, tangent to all three sides. It is the largest circle that fits inside.

The circumscribed circle (or "circumcircle") passes through all three vertices of the triangle. Its center is the circumcenter (where the perpendicular bisectors of the sides meet). It is always larger than the incircle, and for an obtuse triangle the circumcenter lies outside the triangle.

For the 3-4-5 triangle, the circumradius is exactly half the hypotenuse: R = 5/2 = 2.5. So r = 1 and R = 2.5 — the circumcircle is 2.5× the radius of the incircle. The general relationship R ≥ 2r holds for every triangle (the Euler inequality), with equality only for equilateral triangles.

Other inradius formulas worth knowing

• From inradius to incircle area: A_incircle = πr².
• Inradius of an equilateral triangle with side s: r = s/(2√3) = s√3/6.
• Inradius of a right triangle with legs a, b and hypotenuse c: r = (a + b − c)/2. (Try it on 3-4-5: r = (3 + 4 − 5)/2 = 1. Confirms our example.)

Real-world applications

• Largest inscribed object. Cutting the largest possible circular piece from a triangular sheet of metal, wood, or paper — the incircle gives the maximum diameter.
• Triangle centers in design. The incenter is used in CAD when filleting (rounding) the inside of a triangular corner — the fillet's radius cannot exceed the inradius without intersecting another edge.
• Triangle Olympiad problems. A huge fraction of competition geometry problems involve inradius, circumradius, or their relationships (Euler's formula d² = R² − 2Rr where d is the distance between incenter and circumcenter).

Common mistakes

• Using full perimeter instead of semi-perimeter. The formula is r = Area / s, where s = (a+b+c)/2 is HALF the perimeter. Using the full perimeter halves your inradius.
• Confusing incircle and circumcircle. The circumscribed circle passes through the vertices (outside touches inside); the inscribed circle is tangent to the sides (inside touches outside). Easy to swap.
• Forgetting the triangle inequality. If the inputs do not form a valid triangle (one side ≥ sum of the other two), Heron's formula returns 0 or NaN. The inradius is then undefined.
• Mixing units. All three sides must be in the same unit. Inradius comes out in the same unit; incircle area is the squared unit.