12 Circles
Learning Objectives
- Understand the definition of a circle
- Analyze the circumference of a circle
- Calculate the Arc Length
- Analyze the area of a circle
Circle
A circle is the set of all points in a plane that are equidistant (called the radius) from a given point (called the center). An arc is any part of the circle that can be drawn without lifting the pencil.
The Circumference of a circle is the perimeter of the circle. Circumference is measured in the same units as the radius or diameter.
This diagram shows the center, O, the radius, R, the diameter, D, which is two times the radius, and the circumference C.
The ancient Greeks found that if they divided the circumference of any circle by its diameter, they always computed the same number! That number is what we know as [latex]\pi[/latex], pi. There are many approximations of [latex]\pi[/latex], but the most common are [latex]3.14159[/latex] or [latex]\frac{22}{7}[/latex]. These are just approximations of [latex]\pi[/latex]. The number [latex]\pi[/latex] is an irrational number which means it has an infinite number of decimal places that does not repeat.
If we want to use the exact number for [latex]\pi[/latex] you just use the symbol.
Examples
- What is the radius of a circle whose diameter is 22?
- What is the circumference of a circle whose radius is 5? Answer first using the exact value of pi and then using the approximation of 3.14.
The arc length of any arc on the circle can be calculated by using the measure of the central angle that creates the arc, called the angle that subtends the arc and the radius of the circle. A radius of the circle.
A common “name” for an angle is the Greek letter theta, [latex]\theta[/latex]. If we want to find the length of an arc that is a portion of the circle’s circumference, we just need to recognize the proportional reasoning as shown here. The measure of the central anglem as shown in the diagram below, that subtends the arc is [latex]\theta[/latex] and [latex]\theta[/latex] is a portion of the total circumference so the arc length, [latex]L[/latex] can be computed as follows.
The arc length of the arc L in the picture below can be computed as:
[latex]l = \frac{\theta}{360}(2\pi r) = \frac{\pi r \theta}{180}[/latex]
Examples
- What is the arc length if the angle that subtend the arc measures 43° and the radius is 12?
Solution: This is the exact value of the arc length, but to find the approximation you can use either of the approximations for pi noted above.
Note that you should not round until the final calculation and that they are slightly different due to the approximations.
2.
Find the area as an exact value and then using the approximation of [latex]\pi[/latex], [latex]\frac{22}{7}[/latex], of the circle with the diameter [latex]\frac{8}{5}[/latex].
Solution: 
3.
Find the exact value of the area of a circle with diameter [latex]4\pi[/latex]. What is different about this problem?
Solution: 
4. If a circle has an area of 43.1 units squared, find the radius to the nearest tenth.
Solution:
Content created in 2022 by the RSCC Math 1410/1420 OER Team. This work is licensed under a Creative Commons Attribution 4.0 International License.
Circle diagrams from Wikimedia Commons, https://commons.wikimedia.org/w/index.php?search=circles&title=Special:MediaSearch&go=Go&type=image
