Circle
From Thinkmath
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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.
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 area and circumference of a circle:
- Area: A = πr2
- Circumference: C = πd = 2πr
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 represents the ratio of the circumference of a circle to the length of its diameter. Its value is approximately 3.14159265...
Generalizations of the area and circumference formulas can be used to find the length of an arc on a circle or the area of a sector of a circle:
- Sector area:
- Arc length:
In each formula, x represents the measure in degrees of the arc.
Etymology
From Latin circulus, diminutive of circus, meaning 'ring,' perhaps from Indo-European root ker- or sker- meant to turn or bend.
Indo-European ker- also gives rise to Curve, Curvature and crown, coronate, the sun's corona, etc.
Related mathematical terms
The root circum- gives rise to many terms used with mathematical and casual meanings.
Other related words
circulate, circus, circumvent, circumnavigate, circumstance, crown, corona, etc.
