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Inscribed circle of a triangle worksheets

The circle inside a right triangle touches all three sides at once, and its radius comes out of the three side lengths with no area calculation at all: r = (a + b − c) ÷ 2. These free printable worksheets drill it.

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Inscribed circle of a triangle
Inscribed circle of a triangle worksheet — Right triangles, Pythagorean triples, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

The formula looks arbitrary until you walk the tangent lengths. From the right-angle corner, the distance along each leg to where the circle touches is exactly r — that corner and the two touch points make a little square. What is left of the two legs, a − r and b − r, reaches the far corners, and from any corner the two touching lengths are equal, so those two pieces together are exactly the hypotenuse. Rearranged: a + b − c = 2r.

Why this sheet works

Every answer here is a whole number, and that is a property rather than a convenience: a + b − c is even for every Pythagorean triple, so the inradius of a right triangle with whole sides is always whole. The companion sheet asks for the circle through the corners instead, where the same triangle gives a radius that is often a half.

How this one works

One question from this sheet, worked through a step at a time.

The circle inside a 24-32-40 triangle

from the right angle along each leg=r
(24 − r) + (32 − r)=40
(24 + 32 − 40) ÷ 2=8
  1. The circle touches all three sides, so the two pieces of it nearest the right angle make a little square with the two legs: from the corner along each leg is exactly r, the radius.
  2. That leaves 24 − r on one leg and 32 − r on the other. Those two pieces reach the far corners — and from a corner the two touching lengths are equal, so together they are exactly the hypotenuse: (24 − r) + (32 − r) = 40.
  3. Rearranged that is 24 + 32 − 40 = 2r, so r = (24 + 32 − 40) ÷ 2 = 8. The circle inside has a radius of 8 cm — it reaches 16 cm short of one corner and 24 cm short of the other.

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