Arc Length and Area

Arc Length

On a circle of radius 1 an arc of length s subtends an angle whose radian measure is numerically equal to s. That is, s = is the very definition of radian measure. On a circle of radius r we can simply scale the angle measure by a factor of r to obtain arc length. The formula we use is:

Relating s, r and
s = r
If r = 1 then s =

When r = 1 and s = we do not mean they have the same units. Arc length s might be measured in inches while the angle would be in radians. They are the same numerically however. It is probably not surprising that if one circle is twice as big as another then the arc length on the larger circle formed by a central angle will be twice that of the length on the smaller. This relationship is illustrated numerically in this demonstration. The three arc lengths shown should appear to be in the ratios 1:2:3 as the numerical approximations indicate.

s = r

This first exercise gives two of the three values s, r and and asks for the third value. Remember that the formula s = r requires that be expressed in radian measure.

Finding s, r and

Linear and Angular Speed

The linear speed v of a point moving around the circumference of a circle is the rate at which arc length changes with respect to time. When the motion is at a constant rate this can be described as the change in arc length, say in cm, divided by the change in time, say in sec. Thus one possible unit for linear speed is cm/sec. A point traveling at a constant rate that moves once around the circumference of a wheel of radius 1cm in one second has a linear speed of 2 cm/sec since 2 is the arc length traced in one second. Traveling around a wheel of radius 10cm in 4 seconds would give a linear speed of 20/4 = 5 cm/sec.

Angular speed tracks the rate at which the subtended angle changes, not speed around the circumference. For motion at a constant rate this can be described as the change in measure of the subtended angle, say in rad, divided by the change in time, say in sec. Thus one possible unit for angular speed is rad/sec. The first wheel above has an angular speed of 2 rad/sec (once around in 1 second) while the second wheel has an angular speed of 2/4 = /2 rad/sec. We sometimes use revolutions per minute (rpm) in place of angular speed since a revolution is often an easier unit to imagine than a radian. The conversion between the two is quite simple since 1 revolution corresponds to 2 radians. To convert from rpm to rad/min multiply by 2 rad/rev. In the other direction we divide by 2 (that's multiplying by 1/2 rev/rad).

Using s = r we can easily obtain a similar formula relating v and . If we consider s and as changes in arc length and angle measure for a circle rotating at a constant rate we can divide both sides by the corresponding time change to obtain:

Relating v, r and
v = r
If r = 1 then v =

When r = 1 and v = we do not mean they have the same units. Linear speed v might be measured in in/sec while the angular speed could be in rad/sec. They are the same numerically however. In this next exercise, solve for the indicated value and remember that angular speed will use radian measure, not degree measure.

Linear and Angular Speed

Area of a sector

Another useful formula involving the subtended angle calculates the area of a sector of a circle. Start with the formula A = r2, the area of a circle of radius r. The sector formed by the angle contains the fraction /2 of the total area, the ratio of radians to one complete revolution 2 radians. Thus the area of the sector must be A = (/2)x(r2) = (1/2)r2.

Area of a sector
A = r2/2

In this exercise find the area of each sector given r and . Remember to use radian measure for . Round answers to the nearest tenth.

Area of a sector