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
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
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