Radian:
The most natural way to measure the angle θ is not a degree but a radian. It is a two-dimensional plane angle and it is defined as, " the angle (θ) subtended at the center of the circle by an arc with a length equal to the radius of that circle", as shown in the figure and it is equal to the ratio between the lengths of the arc and radius. It is denoted by 'rad'.
Let the angle "θ", which is subtended by an arc of length "S" along a circle of radius "r" then;
θ = `\ frac {S}{r}`(rad) -------(1)
For one revolution.
Length of the boundary of the circle (S) = circumference of the circle = 2Ï€r
by putting values { S = 2Ï€r } in equation (1)
θ = `\frac {2πr}{r}` = 2π rad
This shows that
Numbers of radians in One revolution = 2Ï€ rad
= 2ｘ 3.1416 rad
= 6.2832 rad
There are a little more than 6 radians in one full rotation.
⇒ From equation (1)
S = r θ ---------------(2)
Again using equation (1) when the length of the arc is equal to the length of the radius of that circle i.e. S = r
θ = `\frac {r}{r}` = 1 rad (shown the figure above)
Related Questions:
Relation between degrees and radians
Steradian:
Steradian is a three-dimensional solid angle and is defined as " the angle subtended at the center of the sphere, by which the surface area of the sphere is equal to the square of the radius of that sphere", as shown in the figure below and it is equal to the ratio between the subtended spherical area to the square of the radius. It is denoted by 'sr'.
Angle θ = `\frac {(Spherical'area)}{r^2}` (sr)
But the spherical area = 4Ï€r²
θ = `\frac {4πr^2}{r^2}` (sr)
θ = 4π (sr) ------------ (1)
θ = 4ｘ 3.1416 (sr)
θ = 12.5664 (sr) -------------(2)
Equations (1) and (2) show that there are 4Ï€, or approximately 12.5664, steradians in a complete sphere.
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