Steradian in Solid Geometry

The steradian or square radian is the SI unit of solid angular measure. The symbol used to represent steradian is ‘sr’. It is used to measure two-dimensional angular span in three-dimensional space, similar to the radian which quantifies planar angles.  The term steradian is derived from the Greek word stereos for ‘solid’ and the Latin radius for ‘ beam’.

The solid angle steradian similar to the radian is dimensionless as 1 sr = m².m^-2 = 1. However, It is useful to differentiate dimensionless quantities of different nature. Hence, the steradian symbol (sr) is used whenever appropriate, instead of the derived unit “1” or no unit at all.  For example, the radian intensity can be sated or measured in watts per steradian (W.sr^-1) Foremley, the steradian was considered as an SI supplementary unit, but this categorization was eliminated in 1997 and presently steradian is examined as an SI derived unit.

Definition of Steradian

The steradian definition defines steradian as a solid angle subtended at a center of a sphere of radius r, by a portion of sphere whose area is equivalent to the square of the radius (r²).

The solid angle steradian is related to the portion it cuts out of a sphere.

Solid Angle Steradian (Ω) = \[\frac{A}{r^{2}} sr = \frac{2 \pi rh}{r} sr\]

Where,

• Ω represents the solid angle.

• A represents the area of a spherical cap, 2πrh.

• r represents the radius of the sphere.

• Sr represents the unit steradian.

As the surface area of a sphere is 4πr², the definition states that a sphere measures 4π = 12.56637 sr at its center. By a similar argument, the maximum solid angle (steradian) that can be subtended at any point is 4πsr.

Other Properties of Steradian

If the Area (A) is equal to r², and it correlates to the area of the spherical (A = 2πrh) (where ‘h” represents the height of the cap), then the following relationship holds:

h/r = 1/2πr

(Image will be uploaded soon)

Therefore, 1 steradian resembles to plane angle (that is radian) of the cross-section of simple cone subtending the plane angle 2 θ at angle θ, is equivalent to:

\[\theta = arcos \frac{(r - h)}{r} = arcos (1 - \frac{h}{r}) = arcos (1 - \frac{1}{2 \pi}) \approx 0.572\] rad, or 32.77°.

This angle corresponds to the plane aperture or apex angle of 2 θ ≈ 1.144 rad or 65.54°.

A steradian is also equivalent to the spherical angle of the polygon, retaining an angle access of 1 radian, to 1/4π of a complete square, or (180°/π)² ≈ 3282.80635 square degrees.

The solid angle of a cone whose cross-section subtends the angle two theta (2θ) is

Solid Angle Cone (Ω) = 2π(1 - cosθ) sr

Steradian in Degree

As we can convert radians into degrees, similarly, steradians can be converted into square degrees.

One radian is equal to (180/π) or 57.296 degrees.

One steradian is equals to (180/π)² square degrees or about 3282.8 square degrees.

Steradian Example:

Example

1. You measure the light coming from a powerful drain. Your snore is 70 mm × 70 mm in size, and if you hold the sensor 2 m away it measures 0.2 watts. Calculate the radiant density in W/ sr ( Watts per steradian).

Solution:

One steradian at 2m away, cuts through 2 × 2 = 4m². As the sensor is relatively small in size, its flat surface is approximately the area of the sphere it occupies. Therefore, 0.07 × 0.07 = 0.0049 m².

Therefore, one steradian receives about 0.2 × 4m²/ 0.0049 m² = 163.2.

Facts to Remember

• The value of one steradian is equalled to (180/π)² square degrees or about 3282.8 square degrees.

• Steradian is a measurement unit of solid angle.