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Circumscribed circle of a triangle worksheets

Draw a circle through all three corners of a right triangle and something surprising happens: the hypotenuse turns out to be a diameter. These free printable worksheets ask for the radius, which is then half of one side and nothing else.

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Circumscribed circle of a triangle
Circumscribed 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 reason is the inscribed angle theorem read one step on. The right angle sits on the circle and looks straight across at the hypotenuse; an angle on the circle is half the angle at the center, so the angle at the center is 180° — a straight line through it. A straight line through the center is a diameter. That is the whole proof, and a child who has followed it never has to meet the general formula for a circumradius.

Why this sheet works

So the radius is the hypotenuse halved, and the two shorter sides play no part at all. That is worth saying out loud, because it is the step students distrust: a 15-20-25 triangle and a 7-24-25 triangle are completely different shapes and their circles are exactly the same size.

How this one works

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

The circle through a 7-24-25 triangle's corners

right angle on the circle→180° at the center
the 25 cm side=the diameter
25 ÷ 2=12.5
  1. The corner between the two short sides is a RIGHT angle, and it is sitting on the circle looking straight across at the hypotenuse. An angle on a circle is half the one at the center, so the angle at the center here is 180° — a straight line.
  2. A straight line through the center IS a diameter. So the hypotenuse is the diameter, and the center of the circle is its midpoint — not somewhere inside the triangle.
  3. The radius is half the diameter: 25 ÷ 2 = 12.5. So the circle through all three corners has a radius of 12.5 cm, and the two short sides never come into it.

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