One reason that students get confused about the difference between a circle and a its interior is that we ask them to "Find the circumference and area of the circle," when in fact we mean, "Find the circumference of the circle and the area of the region it encloses." This use of language is common and expected, so teachers just need to be aware of the potential for confusion. An open disk is the set of points in the interior of a circle (but not including the circle) a closed disk is the circle plus all the points inside it see the picture below. To distinguish between a circle and the region it encloses, we can talk about a circle and a disk. Even though the center is required for the definition of the circle, it isn't actually contained in the circle itself. In particular, many students are unsure about whether the center of the circle is part of the circle.
the curve that represents the set of points a fixed distance from the center) and which are just defined by the circle (i.e. It is common for students to get confused about which parts of a figure are part of the circle itself (i.e. This task is best used as a lead-in to the formulas for circumference and area of a circle. The purpose of this task is to help students differentiate between a circle and the region inside of the circle so that they understand what is being measured when the circumference and area are being found.
A circle is the set of points that are a specified distance $r$ from some fixed point $P$ called the center of the circle.