# Linear and Angular Speeds, Area of Sectors, and Length of Arcs

## Introduction to Linear and Angular Speeds

Around the time you’re learning about radian measures, you may have to work with Linear and Angular Speeds, and also Area of Sectors and Lengths of Arcs. Note: For the purposed of this section, we will talk about linear speed and angular speed (scalar quantities), as opposed to linear velocity and angular velocity (vector quantities).

We discussed radians and how they related to degrees of central angles here in the Angles and the Unit Circle section. Before we talk about linear and angular speeds, let’s go over how radians relate back to the circumference of a circle and also revolutions of a circle.

Let’s define some variables to set up the context. $\theta$ is the central angle of rotation in radians, $r$ is the radius of the circle, and $s$ is the arc length, or intercepted arc, which is part of the circumference of the circle. and a revolution (or rotation) is when an object has gone all the way around a circle (or the circle returns back to where it started). These units can all relate back to the circumference of a circle: $2\pi r$, where $r$ is the radius. (Remember in a Unit Circle, the circumference is just $2\pi$, since $r=1$). Makes sense, right?

Note that the words radian and radius are related, since there are $2\pi r$ (radians) in a revolution, and $2\pi r$ (radius) is the measurement of the circumference.

One of the most important concepts is that the length of an intercepted arc (part of the circle’s circumference) is the radius times the radian measure of that arc’s central angle. To see this, set up a proportion comparing this arc to the whole circumference:

$\begin{array}{c}\displaystyle \frac{{\text{Length of arc}}}{{\text{Circumference}}}=\\\displaystyle \frac{{\text{Arc }\!\!’\!\!\text{ s Angle}}}{{\text{Total Angle Measurement in Circle}}}\,\,\\\,\displaystyle \frac{s}{{\cancel{{2\pi }}r}}=\displaystyle \frac{\theta }{{\cancel{{2\pi }}}}\,\,\,\,\,\,\,\,\,\text{Thus, }s=r\theta \end{array}$

Linear speed is the speed at which a point on the outside of the object travels in its circular path around the center of that object, described in units like miles per hour, meters per second, and so on. Angular speed is the rate at which the object turns, described in units like revolutions per minute, degrees per second, radians per hour, and so on. Both will be discussed in more detail below.

A rule of thumb is that you typically use a radius with linear speed, and radians with angular speed.

Below are the linear and angular speed formulas. Note: It’s good to know these formulas, but when I do most of the problems, I just use Unit Multipliers (Dimensional Analysis) to get to the units needed for the answer of the problem!!

## Linear Speed

Linear speed is the speed at which a point on the outside of the object travels in its circular path around the center of that object. The units can be any usual speed units, such as miles per hour, meters per second, and so on.

Remember the equation $\text{Distance}=\text{Rate}\times \text{Time}$, or $\displaystyle \text{Rate (Speed)}=\frac{{\text{Distance}}}{{\text{Time}}}$, to think about how fast an object along the circumference of a circle is changing.

Think of a car that drives around in a circle on a track with arc length $s$ (the actual length of the curvy part – part of the circumference). Based on the formula above, the formula for the speed around a circle, or the linear speed, is $\displaystyle v=\frac{s}{t}$, where $s$ is the arc length and $t$ is the time. Note that linear speed needs to have a circumference (or radius) in the problem!

Here are some typical problems. Note that we have to use Unit Multipliers (Dimensional Analysis) when the units don’t match.

## Angular Speed

Angular speed is the rate at which the object turns, described in units like revolutions per minute, degrees per second, radians per hour, and so on. Angular speed has to do with how fast the central angle of a circle is changing, as opposed to the circumference of the circle.

Again, think of a car that drives around in a circle on a track with central angle $\theta$. The formula for the speed around a circle in terms of this angle, or the angular speed is $\displaystyle \omega =\frac{\theta }{t}$, where $\theta$ is in radians, and $t$ is the time.

Note that angular speed does NOT need a circumference (or radius) in the problem! Also, radians have no units!

Here are some typical problems. Note that we have to use Unit Multipliers (Dimensional Analysis) again when the units don’t match.

Here’s an example that shows the difference between finding angular speeds and linear speeds. Notice with angular speeds, we ignore the radius, since we are just dealing with rotation of an angle. Also notice that:

$\text{Linear Speed}=\text{Radius}\times \text{Angular Speed}$, or $\displaystyle \text{Radius}=\frac{{\text{Linear Speed}}}{{\text{Angular Speed}}}$

Here are more problems with linear and angular speeds, and rotations/revolutions:

## Area of Sectors

Let’s get the area of a sector of a circle based on the radius and a central angle in radians. We know from geometry that a sector of a circle is like a pizza slice; it’s a region bounded by a central angle and its intercepted arc.

You may have seen in geometry how to use proportions to get the area of a sector based on radius and degrees of the sector: $\displaystyle \frac{{\text{area of the sector}}}{{\text{area of the whole circle}}}=\frac{{\text{degrees of the sector}}}{{360{}^\circ }}$.

Now get the area of a sector based on a radius and a central angle. Do you see how this proportion gives us the ratio of that particular part of the circle?

$\displaystyle \frac{{\text{area of sector}}}{{\text{area of whole circle}}}=$

$\displaystyle \frac{{\text{arc length of sector}}}{{\text{arc length of whole circle (circumference)}}}$

Use the fact that the area of a circle is $\pi {{r}^{2}}$, the arc length of a sector is $r\theta$, and the arc length of a whole circle is $2\pi r$. Solve for the area of a sector given a radius and central angle: $\displaystyle \frac{{\text{area of the sector }(A)}}{{\pi\,{{r}^{2}}}}=\frac{{r\theta }}{{2\pi r }};\,\,\,\,\,\,A=\frac{1}{2}{{r}^{2}}\theta$. Problems are below.

## Length of Arcs

Again, with proportions, we have:

$\displaystyle \frac{{\text{length of intercepted arc }(s)}}{{\text{ arc length of whole circle (circumference}=2\pi r\text{)}}}=$
$\displaystyle \frac{{\text{radians of central angle of intercepted arc (}\theta \text{)}}}{{2\pi }}$.

Cross multiply to get $s=r\theta$, where $s=$ the length of the intercepted arc, $r$ is the radius of the circle, and $\theta$ is the intercepted arc. This the formula we need!

Here are some problems; get the area of a sector and length of intercepted arc, given the radius and central angle:

Understand these problems, and practice, practice, practice!

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On to Graphs of Trig Functions – you’re ready!

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