Arc Length (s) versus Radius (r) - Constant of Proportionality
Find the arc length (s) when [math]\theta=180^\circ[/math] and [math]r=1[/math], [math]r=2,[/math], [math]r=3[/math], [math]r=4[/math], and [math]r=5[/math].[br][br][img]data:image/png;base64,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[/img]
[size=150][size=200]Below is the [b]s vs. r[/b] graph ([u]arc length vs. radius[/u]) for radius values of 1, 2, 3, 4 and 5. [/size][/size]
Plotting s vs r for r=1, 2, 3, 4, 5 gives the following graph:
Consider the formula for arc length (s) in terms of radius (r).[br][br][img]data:image/png;base64,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[/img][br][br]Plotting s vs r with for every angle gives a linear relationship. What is the slope of this line?
Adjust the angle slider. How does the slope of the line change for different angle measures? At higher angle measures, zoom out to see all the plotted points.
The arc length (s) and the radius (r) are directly proportional, meaning they increase and decrease at the same rate. The ratio between two directly proportional quantities is called the [b]constant of proportionality[/b]. The constant of proportionality for s / r is the slope of the line (graphed above) or [math]\frac{s}{r}=\frac{\theta}{360}\cdot2\pi[/math]. [br][br]This is how the radian measure for an angle is defined, where [math]\theta\left(radians\right)=\frac{\theta\left(degrees\right)}{360}\cdot2\pi[/math]. Simplifying this gives a formula for finding the radian measure of any angle. Which is the correct formula?
Use the formula for radians, [math]radians=\theta\left(degrees\right)\cdot\frac{\pi}{180}[/math], to select all of the correct radian measures below.
Radian Measure - Definition
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[/img][/size][br][br][i][size=100][size=150][size=200]Where, [br]s = arc length[br]r = radius[br][/size][/size][/size][/i]
Rearrange the arc length formula to find an equation for s/r. [br][br]This is the radian measure, defined as the ratio of the arc length to the radius.
Adjust the angle slider to change the angle measure. What do you notice about the radian measure for each angle?
Adjust the angle measure. What is true about the value of the arc length (s) divided by the radius (r)?
How does this relate to circle similarity?