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trigonometry

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if a circle has a radius of 8 centimeters, what is the exact length of the arc intercepted by a central angle of 90 degrees.

May 15, 2018

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An accompanying diagram is an asset geometrists use to condense large amounts of information into something more manageable and visual. I have created a reference that suits this situation perfectly. Notice the features of this diagram and how it relates to the actual problem:

• The circle has a radius of 8 centimeters
• There is a central angle of 90°
• The intercepted arc, $$\widehat{BC}$$, is present

Now that the important features are obvious and present, we can actually move on to the solving bit.

One elementary fact about a circle is that its entire circumference can be represented with this formula:

$$\text{Circumference}=2\pi r$$

However, there is a slight issue with this formula for this particular context: The problem only asks for a certain portion of the circumference--not the whole thing. How can we remedy this issue? Well, let's just pretend for a moment that we actually knew that certain percentage, say 20%.

If we knew that the problem only wanted 20% of the circumference, then we would just multiply the circumference by that arbitrary percentage, 20 in this case. The formula would be $$\text{Circumference}=0.2*2\pi r= 0.4\pi r$$

Ok, we are making some great progress on this problem! We still need a way of determining that percentage, though. I will introduce another well-known fact about circles: They have 360 degrees in total. We also know the central angle measure, which is 90°. This means that 90 out of the 360 degrees encompass the subtended (also referred to as intercepted) portion. This means that the "certain percentage" is determined by the central angle measure. We can generalize this and create a formula for this!

 $$\text{Arc Length}=2\pi r*\frac{\theta}{360}$$ This is what we have determined up to now. $$\theta$$ is generally the Greek letter used to denote a variable in a situation similar in nature to how it is in this problem. There is only one bit of simplification possible here. $$r=8;\theta=90\\ \text{Arc Length}=\frac{\pi r\theta}{180}$$ Substitute in the values and solve. $$\widehat{BC}=\frac{8*90*\pi}{180}=4\pi\text{cm}$$ You are finished!
May 15, 2018