Incenter
In short: A triangle’s incenter is the point where its three angle bisectors cross, the same distance from every side, and the hub of the largest circle that fits inside.
GeometryGeometryG-C.A.3
What is Incenter?
Split each of a triangle’s three corners into two equal halves and the three splitting rays pass through one common point, the incenter. Any spot on such a ray is equally far from the two sides of its corner, so the crossing point is equally far from all three sides.
That shared distance is the inradius \(r\). A circle of radius \(r\) around the incenter kisses each side exactly once and is called the inscribed circle. The incenter is always inside the triangle, and \(r = \frac{\text{Area}}{s}\), where \(s\) is half the perimeter.
How to spot it
- A problem asks for the largest circular tabletop, plate or pizza that fits inside a triangular space.
- The words inscribed circle, inradius or all three angle bisectors appear.
- A point must be equally far from three sides of a triangle rather than from three corners.
- The sides of a triangle are given and the radius of the circle inside is requested.
The method
- Compute the area of the triangle and half its perimeter \(s\).
- Divide: the inradius is \(r = \frac{\text{Area}}{s}\).
- To locate the center, bisect two corners and find where the bisectors cross.
- Verify that the distance from the center to every side equals \(r\).
Incenter: solved example
Problem
A woodworker cuts a triangular tray with edges 9 in, 12 in and 15 in, and wants the biggest round coaster that fits inside. What radius should it have?
Solution
Because \(9^2 + 12^2 = 15^2\), the tray is a right triangle with area \(\frac{1}{2}(9)(12) = 54\) square inches. Half the perimeter is \(s = \frac{9 + 12 + 15}{2} = 18\).
Then \(r = \frac{54}{18} = 3\). Placing the legs on the axes, the center is \((3, 3)\), and its distance to the slanted edge \(3x + 4y = 36\) is \(\frac{|9 + 12 – 36|}{5} = 3\). The coaster radius is 3 inches.
Try it
Question 1
A triangle with sides 13, 14 and 15 has area 84. Find the radius of its inside circle.
Show the answer
\(s = \frac{13 + 14 + 15}{2} = 21\), so \(r = \frac{84}{21} = 4\).
Question 2
Could the point where the three angle bisectors meet fall outside an obtuse triangle?
Show the answer
No. Each bisector runs inside the triangle from its corner, so their crossing is always inside.
Incenter vs. Circumcenter
The incenter belongs to angle bisectors, reaches the three sides equally and anchors a circle tucked inside the triangle. The circumcenter belongs to perpendicular bisectors, reaches the three corners equally and anchors a circle drawn around the triangle. Inside versus around is the quickest test.
Keep going
Lessons
Worksheets and quizzes
- Geometric Constructions – Worksheet
- Geometric Constructions – Quiz
- Theorems About Triangles – Worksheet
- Theorems About Triangles – Quiz
Related words
Whole dictionary: A–Z · Geometry · Geometry