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The quantities which have the same dimensions as those of solid angle are:
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strain and angle
Solid angle $d \Omega=\frac{d A}{r^2}$ has dimensions $\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}^0\right]$
Strain $=\frac{\Delta l}{l}$ has dimensions $\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}^0\right]$
Angle measured in radians is also dimensionless $\left[\mathrm{M}^0 \mathrm{~L}^{\circ} \mathrm{T}^0\right]$
$\theta=\frac{I}{r}$
Strain $=\frac{\Delta l}{l}$ has dimensions $\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}^0\right]$
Angle measured in radians is also dimensionless $\left[\mathrm{M}^0 \mathrm{~L}^{\circ} \mathrm{T}^0\right]$
$\theta=\frac{I}{r}$
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