The quantities which have the same dimensions as those of solid angle are :

The solid angle (Ω) is defined as the area A subtended on a unit sphere divided by the square of the radius r:

$\Omega \; =\; \backslash frac\{A\}\{r^\{2\}\}$

Since both the numerator (area) and the denominator (r²) have dimensions of $L^\{2\}$, the ratio is dimensionless. Therefore, the solid angle has no physical dimensions.

To find other quantities with the same dimensions, we look for dimensionless measures.

1. Strain is the relative change in length:

$\epsilon \; =\; \backslash frac\{\Delta L\}\{L\}$

Both ΔL and L have dimensions of length ($L$), so their ratio is also dimensionless.

2. Angle (expressed in radians) is the arc length s divided by the radius r:

$\theta \; =\; \backslash frac\{s\}\{r\}$

Again, both s and r are lengths, giving a dimensionless quantity.

Other options involve quantities with dimensions:

• Stress has dimensions of pressure ($M·L^\{-1\}·T^\{-2\}$).
• Arc length is a length ($L$).
• Angular speed has dimensions of inverse time ($T^\{-1\}$).

Hence, the only pair that shares the dimensionless nature of a solid angle is **strain and angle**.