Inscribed Circle — Definition, Formula & Examples

Inscribed Circle
Incircle

The largest possible circle that can be drawn interior to a plane figure. For a polygon, a circle is not actually inscribed unless each side of the polygon is tangent to the circle.

Note: All triangles have inscribed circles, and so do all regular polygons. Most other polygons do not have inscribed circles. For a regular polygon, the inradius (the radius of the inscribed circle) is called the apothem.

Inscribed Circle of a Triangle
Triangle with incenter marked at intersection of three angle bisectors, inscribed circle shown with three inradius segments...

Key Formula

r = \frac{A}{s}

Where:

$r$ = Inradius — the radius of the inscribed circle
$A$ = Area of the triangle
$s$ = Semi-perimeter of the triangle, where s = (a + b + c) / 2 and a, b, c are the side lengths

Worked Example

Problem: Find the radius of the inscribed circle of a triangle with sides 6, 8, and 10.

Step 1: Compute the semi-perimeter by adding all three sides and dividing by 2.

s = \frac{6 + 8 + 10}{2} = 12

Step 2: Find the area of the triangle. Since 6, 8, 10 is a right triangle (6² + 8² = 10²), the area is half the product of the two legs.

A = \frac{1}{2}(6)(8) = 24

Step 3: Apply the inradius formula by dividing the area by the semi-perimeter.

r = \frac{A}{s} = \frac{24}{12} = 2

Answer: The inscribed circle has a radius of 2 units.

Another Example

This example uses a regular polygon instead of a triangle, showing the apothem formula for inscribed circles in regular polygons.

Problem: Find the radius of the inscribed circle of a regular hexagon with side length 6.

Step 1: Recall that for a regular polygon, the inradius (apothem) can be found using the formula involving the side length and the number of sides.

r = \frac{s}{2\tan\!\left(\frac{\pi}{n}\right)}

Step 2: Substitute n = 6 (hexagon) and s = 6 into the formula.

r = \frac{6}{2\tan\!\left(\frac{\pi}{6}\right)}

Step 3: Evaluate the tangent. We know tan(π/6) = 1/√3.

r = \frac{6}{2 \cdot \frac{1}{\sqrt{3}}} = \frac{6}{\frac{2}{\sqrt{3}}} = \frac{6\sqrt{3}}{2} = 3\sqrt{3}

Step 4: Compute the decimal approximation.

r = 3\sqrt{3} \approx 5.196

Answer: The inscribed circle of the regular hexagon has a radius of 3√3 ≈ 5.196 units.

Frequently Asked Questions

What is the difference between an inscribed circle and a circumscribed circle?

An inscribed circle (incircle) is the largest circle that fits inside a polygon, tangent to every side. A circumscribed circle (circumcircle) is the smallest circle that passes through every vertex of the polygon, containing the entire figure. For a triangle, the incircle center is found using angle bisectors, while the circumcircle center is found using perpendicular bisectors of the sides.

Do all polygons have inscribed circles?

No. All triangles have inscribed circles, and so do all regular polygons. However, most irregular polygons with four or more sides do not have an inscribed circle. A polygon has an inscribed circle only if there exists a single circle tangent to every one of its sides simultaneously; this is a special geometric condition that most shapes fail to satisfy.

How do you find the center of an inscribed circle?

For a triangle, the center of the inscribed circle (called the incenter) is the point where all three angle bisectors meet. This point is equidistant from all three sides of the triangle, and that common distance is the inradius. For a regular polygon, the incenter is simply the center of the polygon.

Inscribed Circle (Incircle) vs. Circumscribed Circle (Circumcircle)

	Inscribed Circle (Incircle)	Circumscribed Circle (Circumcircle)
Definition	Largest circle fitting inside the polygon, tangent to every side	Smallest circle passing through every vertex of the polygon
Radius formula (triangle)	r = A / s (area ÷ semi-perimeter)	R = abc / (4A) (product of sides ÷ 4 × area)
Center found by	Intersection of angle bisectors (incenter)	Intersection of perpendicular bisectors (circumcenter)
Exists for all triangles?	Yes	Yes
Exists for all polygons?	No — only triangles and regular polygons (plus special cases)	No — only cyclic polygons

Why It Matters

You encounter inscribed circles in geometry courses when studying triangle centers, angle bisectors, and the relationships between a shape's area and perimeter. The incircle also appears in competition mathematics and real-world design problems where you need the largest circular object that fits inside a given boundary, such as fitting a round table inside a triangular room or cutting the largest circular piece from a triangular sheet of material.

Common Mistakes

Mistake: Confusing the inscribed circle with the circumscribed circle and using the wrong formula.

Correction: The inscribed circle is inside the polygon and tangent to its sides (use r = A/s for triangles). The circumscribed circle is outside or around the polygon, passing through its vertices (use R = abc/(4A) for triangles). Draw a sketch to confirm which circle the problem asks for.

Mistake: Forgetting to use the semi-perimeter and instead dividing the area by the full perimeter.

Correction: The formula is r = A/s, where s is the semi-perimeter (half the perimeter), not the perimeter itself. Using the full perimeter gives an answer that is exactly half the correct inradius.

Related Terms

Circumcircle — Circle through all vertices, counterpart to incircle
Incenter — Center of the inscribed circle of a triangle
Inradius — Radius of the inscribed circle
Apothem — Inradius of a regular polygon's inscribed circle
Tangent Line — Each side of the polygon is tangent to the incircle
Regular Polygon — Always has an inscribed circle
Centers of a Triangle — Incenter is one of four classic triangle centers
Circle — The fundamental shape of the incircle