Arc Lengths

Arc lengths:
Let us talk about arc length for grade IX students. An arc length is the length of the part of the circumference of any circle. For measuring arc length, we will first understand the concept of radian.
The equation of radian is:
l=rθ
here, l represents the length of the arc.
Every arc will make an angle θ at the center of the circle whether it is large or very small.
The ratio of the circumference to the diameter of the same circle represents the radian.
In mathematical form,
∏=circumference/diameter      ------------- (1)
Now, as we know that
diameter=2*radius
putting In equation 1, we get
∏=circumference/2*radius.
Or, 2∏=circumference/radius
It is one revolution of a circle. One revolution of a circle can be divided into four right angles. The circumference covers these four right angles. (know more about cbse board, here)
Here,
∏ stands for radian measure for 1 revolution.
The arc length formula is:
θ =l/r,
where, l=arc length
r=radius of circle.
radian measure(θ)=l/r
The ratio l/r is a real number and it is calculated as an angle (θ) at the center. The angle (θ) is measured in radians.
Let us discuss a theorem below,
Theorem: if the ratio of the arc length to the radius of circle is same for the two circles, then they must possess same angle θ at the center.
In mathematical terms,
if l1/r1=l2/r2
then,
θ1=θ2
Let us take an example to calculate arc length.
Example 1: calculate the arc length of a circle if it is given that the radius is 10 cm and central angle is 2.35 radians.
Solution:
As we have discussed earlier,
θ=l/r
l is the arc length of the circle
hence,
l=θ r
θ=2.35 radians,
θ is already in radians.
If it is given in degrees, we will have to first change it into radians.
L=2.35*10=23.5 cm.
Let us discuss another example based on the theorem above.
Example 2: Two circles have the same angle θ at the center created by their arcs.
It is given that r1=20 cm
r2=15 cm
l1=25 cm
l2=?
Solution:
we have to calculate the length of the arc of the second circle.
As given θ1=θ2=θ,
using the theorem,
l1/r1=l2/r2
it implies,
l2=(l1/r1)*r2
l2=(25/20)*15
l2=18.75 cm
In the next session we will discuss about different properties to solve problems in Grade IX

time and angle measures

Time and angle measures are one of the important types of mathematical problems. We all know that time problems (clock angle measure problem) are all about finding the angles between the hands (Hands are minute hand or hour hand or second hand) of an analog clock. This is very important part of geometry. In geometry the measurement unit for angle is the “Degree”. A circle is divided into 360 equal degrees.  Moreover, degrees are divided into two terms, known as Minutes and Seconds. This division of degree is not universal. (To get help on central board of secondary education books click here)
These problems generally relate to two different measures which are angles and time. For such problems we always consider the change rate of the angle in degree per minute (deg/min) and we may also use arc lengths in this. In an analog clock the hour hand of the clock turns 360 degree in 12 hours which are 720 minutes or we can say 0.5 degree per minute. In the same analogue clock the minute hand turns 6 degree per minute or 360 deg in 1 hour (60 minutes).
Now to solve related problems we need to apply some formulas in form of general equations. These equations are:
1.      Equation for angle of the hour hand of the clock:
Ahr = ½ Me = ½ ( 60 H + M)
Here Ahr is the angle of the hand measured clockwise from the exact time 12’o clock. M is the minute past hour and Me is minute past 12’o clock.
2.      Equation for degree on minute hand of the clock:
Amin  = 6*M
Where Amin is the angle of the hand measured clockwise from the time 12’o clock and its unit is degree.
For example: Time given is 5:30 then the angle of hour hand in degrees:
Ahr  = ½(60*5 + 30) =165 degree
Similarly for the same time the angle of minute hand in degree:
Amin  = 6*30  = 180 degree.
So today we learnt about Times and Angle measures. In the next session I will tell you about arc lengths and In the next session we will discuss about Rate, distance in geometry.