For me, in Geometry this year, of all topics, circles were the one that I seemed to grasp most. Something about them just made sense, perhaps that every concept involving circles seems to tie back to a central point (pun intended).
Thinking about a circle on a simple level, a circle is made up of 360 degrees (which made understanding arcs a piece of cake), its circumference is 2πr, the area is πr^2, and every other aspect essentially builds upon these ideas. Given these, you can determine the values of interior angles built into the circle, find areas of inscribed/circumscribed shapes, and solve for almost any related missing aspect. We’ll try and get into a few of these today.
In simple form, it’s often best to think of a pizza, made up of several (hopefully) evenly cut slices. Imagine a center point in the middle of the circle where the ends of each slices meet. If the pizza is evenly divided, then the angle formed by the ends of each slice should be equal, and in total, add up to 360 degrees.
For example, if you had a pizza divided up into 8 slices, each slice would have an angle of 45 degrees. This is what we call a central angle. The crust of each slice forms an arc, which similarly, adds up to 360 degrees. Therefore, the measure of the arc (different than the length of an arc, which is related to circumference), will always be the same as the relative interior angle, and vice versa.
Each side of the pizza represents the radius of the circle, which is half of the diameter. That information, especially the radius, is vital in substituting values in solving for various solutions. For example, one way this information can be utilized is finding the area of a sector. The area of a sector can be made of of multiple arcs, or in our example, multiple slices of pizza.
Here’s the formula you’ll want to know:
Measure of defined arc measure / 360 = area of sector / area of circle
In solving something like this, you’ll be given three pieces of information, and whatever you don’t know can be substituted with the variable x.
For me, it always made sense that area of a sector tied in with the total area, whereas in this case, the length of an arc ties in with the circumference (or the “total length”). To find the length of an arc, this is the formula:
Measure of defined arc measure / 360 = length of arc / circumference
Remember- if the defined arc measure is not given, refer to the measure of the interior angle. This is important though: IF the angle is not tied to a center point, it is instead defined as an inscribed angle. In that case, the measure of the angle is 1/2 that of the arc measure the endpoints of each line of the angle are tied to, and reversed, the arc measure is double.
Given this diagram, let’s try each of these formulas out. First, we’ll find the area of arc CEF (which includes everything in between, like arc DE). Here is an outline of how I would solve such a problem:
- We know that <BGA (65 degrees given its arc measure) is a vertical angle with <EGF, so their arc measures must be the same.
- <CHD is an inscribed angle, so the measure of arc CD is 84 degrees.
- In order for all arc measures to add up to 360 degrees, the measure of arc DE must be 14 degrees. (Keep in mind that even though it is not drawn, angle <DGE, a central angle as properly defined, still exists).
- Filling in the aspects of our formula, then, would bring us to this:
163 degrees / 360 = area of sector (x) / π * 12^2
Set up vertically, you can cross multiply and using algebra, solve for x. You should get 204.83 square units, if rounded to the nearest hundredth. Now, imagine that all of the arc measures given are instead arc lengths (so the length of arc CD is 84, EF is 65). Using the same formula, finding the length of the same arc would be solved like this:
163 units / 360 = length of arc CEF (x) / 2 * π * 12
My answer was 34.14 units.
And again, using algebra, you don’t just have to find the areas of sectors or lengths of arcs. You can plug in whatever information you have and solve for the missing variable.
Like I said, there are infinite possibilities that can come from working with circles. In the next edition, I’ll tackle finding the areas of shapes OTHER than circles, using properties of circles, along with some other basics of area. Stay tuned!
Thank you for reading!