Shape: Circle

Topics: Mathematical Language

Mathematical Domain: Geometry, Measurement & Data

Meaning

Informal sense: “Circle” is a familiar word to most children very early, but with little more than an informal sense of “round.” We sit “in a circle” without much thinking about how round it is, and without picturing the geometrical object associated with the word.

As students learn geometry, we help them refine the meaning from the informal sense — the shape of a ring, wheel, or frisbee — and take on a more formal mathematical meaning. For one thing, the circle is a two-dimensional figure: that is, a photograph of a ball may be circular, but the ball, itself, is not (a ball is three-dimensional). And “round” is not enough: all of the figures below are “round” but for something to be a circle, every point on it must be the same distance from the center.

Some points on the ellipse and egg-shape are farther from the center (red) than others; all points on the circle are the same distance from the center.

Formal definition: A circle is the set of points in a plane that are equidistant from a fixed point (the center of the circle).Note that the formal definition excludes the “inside” of the circle: those points are closer to the center than the points on the circle itself. When students are asked to point to a circle, they often point inside the circle, and may refer to the circular “rim” of this shape as the circumference. In precise usage, the closed curve itself is the circle, while the term circumference refers to the length of the curve. (The word “disk” is used to refer to the region enclosed by the circular boundary. But even mathematicians may say “cut the circle in half” instead of “cut the disk in half” when context makes the meaning clear enough.)

One way to clarify the language is to present a picture in which the circle and its interior are colored differently. In this picture, the circle is red; its interior is yellow. The exterior of the circle is the entire remaining plane, not on or enclosed by the circle. This picture shows a small square region of that plane, and colors the exterior of the circle blue.

Measurements on circles

Two important formulas can be used to find the circumference and area of a circle:

Circumference: C = πd = 2πr
Area: A = πr²

In each formula r represents the length of the radius of the circle, while d represents the length of the diameter, or twice the length of the radius. The number pi (π) is an irrational number that is defined as the ratio C/d — the ratio of the circumference of a circle to the length of its diameter — though precise computations of the value of π are never based on measurements. Its value is approximately 3.14159265…

Discovering the circumference formula

Because $\pi = \frac C d$ is the definition of pi, the “formula” for the circumference of a circle is really nothing new, just a restatement, or re-use, of the definition of pi.

If $\pi = \frac C d$ , then $\pi \cdot d = C$ .

Discovering the area formula

We cannot measure the area of a circle precisely by placing squares on it; they won’t fit, no matter what size we choose. So, to find a formula for the area of a circle, we “dissect” it, cut it into pieces whose areas we can measure. (See Area formulas for examples of dissecting parallelograms and triangles.)

Sectors of a circle look a bit like triangles, especially if we subdivide the circle into enough of them. If we dissect the circle into eighths (left hand circle in the figure below), the difference (shown in pink) between a complete sector and the triangle it contains (shown in blue) is pretty substantial. We could compute the area of that blue triangle, multiply by 8, and then just be satisfied with that underestimate of the area of the circle.

But we could do better. If we dissect the circle into twelfths (center, above) the error — the amount of pink left over — is considerably less. And if we dissect the circle into 24ths (right, above), the leftover is smaller yet, even accounting for the fact that now there are 24 such left-over pieces.

More importantly, we notice something else. As the circle is dissected into more sectors — and the distinction between “sector” and “triangle” evaporates more and more — the dimensions of each triangle are now easier to relate to the dimensions of the circle. If we’ve dissected the circle into 24 sectors, the short side of the triangle is (almost) 1/24 of the circumference of the circle, and the height of that triangle is (almost) the radius of the circle. So, using these approximations, the area of the triangle is

$A_{triangle} = \frac 1 2 b h = \frac 1 2 \cdot \frac 1 {24} C \cdot r$ But there are 24 of those triangles in the circle, so the area of the circle would be

$A_{circle} = 24 \cdot \frac 1 2 \cdot \frac 1 {24} C \cdot r = \frac 1 2 \cdot C \cdot r$ and we know that C = 2πr, so

$A_{circle} = \frac 1 2 \cdot C \cdot r = \frac 1 2 \cdot 2\pi r \cdot r = \pi r^2$ One more bit of care. Our final formula pretended that the base and height of the triangle were exactly what we wanted — exactly $\frac{1}{24}$ of the circumference and exactly the radius — which we knew was not true. What saves the reasoning is that we can make the approximation as accurate as we like. If we had dissected the circle into 1000 sectors — totally pointless to draw — the base and height of the triangle would be even closer to being 1/1000 of the circumference of the circle and precisely the radius. Since the reasoning always produces the same result, and we can get the approximation that supports that reasoning as accurate as we like — cut the circle into a zillion sectors, or a zillion zillion sectors — the result is general.

Another image that is sometimes used to get a feeling for the area of the circle is shown below. There, all the sectors are spread apart like wedges of an orange.