Triangle circlesThe circle through the corners
Name: Date:
Each circle passes through all three corners. Find its radius.
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6 cm
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.
If they get stuck
Circumscribed circle of a triangle
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 a sheet like this one, worked a step at a time — with a sentence saying what just happened and why.
Circumscribed circle of a triangle, step by step