What is a circle? In this lesson, find out all about the circle and its many parts, including circular arcs and semicircles. Also, discover how a locus works in creating a circle, parallel lines and more.
The Sum of All Dots
Did you ever look at the stars and think 'Yep, I totally see a lion, a ram and a seated woman'? Because other than the Big Dipper, I usually just see a bunch of dots. I want those dots to form more complete pictures. When we talk about a few dots becoming so great in number that an actual line is formed, we're talking about a locus.
A locus is an invasive insect species that destroys crops and can be summoned by Moses. Wait, no, that's 'locusts' with a 't.' Totally different thing.
A locus is a set of points that share a common property.
Let's consider a few examples. If we look at all the points, or the locus of points, that are d distance from a line above or below it, we get parallel lines - not just one or two stars, but so many that it's an actual line.
The locus of points that are equidistant from two parallel lines is a line midway between them and parallel to both. It's like a center line on a perfectly straight road. If we take two points and look at the locus of points that are equidistant from those points, we get a line that bisects the line connecting our two points. It's also perpendicular to that line.
Now those were all boring straight lines. What if we get a bit more interesting? Let's say we look at a few points that are all a fixed distance, d, from a point, P. That looks like a constellation. And what if we connect the dots by looking at all of the points, or the locus of points, that are d distance from point P? We get a circle!
Points in a circle
That's really all a circle is. It's just the set of all points that are the same distance from a central point. We call that central point the center. Creative, I know.
But a circle has many other cool parts to know that have more interesting names. First, we call that distance from the center the radius. That can be any line from the center to the edge.
Radius- Center point to edge of circle
If we go from one edge of our curve to the other, passing through the center, that's a diameter. A diameter, then, is just two connected radius lines. To remember this, think about how the word 'radius' is shorter than the word 'diameter,' so it's the shorter line.
Diameter- Two connected radius lines
If we have a square or a rectangle, we call the distance around the edge the perimeter. Circles, though, are special. They get their very own word for perimeter. We call the distance around the circle the circumference. This word comes from the Latin (of course) and means 'to carry around.'
Notice that the words circle and circumference both begin with 'circ' - as would a three-ring circus, which is also very circle-focused. On a pizza, the circumference is the handy crust that keeps our fingers from getting messy.
If we're just looking at part of the circumference, we call that a circular arc. That's like the crust on just one pizza slice. In this circle, if we label our points, we could call this arc AB.
But wait - on our circle, below, couldn't arc AB be shown like this? That's where we get major arcs and minor arcs. When we have two points like this, the major arc is just the longer arc connecting the two points. The minor arc is the shorter one.
Longer arc- Major Arc
Finally, if our circle is a pizza and we eat half the pizza, then, well, we've eaten a lot of pizza. Also, the shape that's left? That's called a semicircle, which is just half a circle. This is neither a major nor a minor arc because the arc lengths are equal.
In this lesson, we discussed a locus, or a set of points that share a common property. We looked at different examples of what a locus of points can be. One of those is a circle. This is the set of all points the same distance from a center.
We then covered many of the parts of circles, including the radius and the diameter. We learned about the circumference, or the circle's perimeter.
Then we learned about circular arcs. A circular arc is a part of a circle's circumference. We differentiated between the shorter arc connecting two points, which is the minor arc, and the longer one, or major arc.
After completing this lesson, you should be able to:
- Identify a circle
- Name the parts of a circle
- Describe circular arcs and the differences between minor and major arcs