Solid Angle Diagram: What is a Plane & Solid Angle? 1 sr & 1 rad Defined

[SpaceExplorer]

What is a plane angle and a solid angle?(please provide diagram if possible) Also, what is the definition of 1 sr and 1 rad?

 

[mathman]

What is a plane angle and a solid angle?(please provide diagram if possible) Also, what is the definition of 1 sr and 1 rad?

Sorry - no diagrams.

A plane angle is defined as the angle between two straight lines. To get the units, consider a circle of radius 1. The circumference is of length 2pi. The angle (radians) between two radii is defined by the length of the arc defined by the two radii.

A solid angle is defined in analogous way using a sphere of radius 1. The sphere has a surface area of 4pi. A solid angle (sphereradians) is defined by the area of a figure on the surface of the sphere. The solid angle is given as the angle from the center of the sphere, where the surface figure is traced back to the center. I know that this sounds clumsy, but think of it by starting with a spherical triangle, leading to a wedge shaped figure.


Note - this is a math question. It is not physics.

 

[tiny-tim]

Welcome to PF!

What is a plane angle and a solid angle?(please provide diagram if possible) Also, what is the definition of 1 sr and 1 rad?

Hi SpaceExplorer ! Welcome to PF!

I'll just add this to what mathman says:

Solid angle is like an area in your field of vision.

When a star moves across the sky, you measure that as an angle, of so many radians, as a fraction of 2π.

But you measure the "area" of the moon (or of the field of vision of a telscope) as a solid angle, of so many steradians, as a fraction of 4π.

 

[Priya123]

1 sr means 1 steradian steradian is the SI unit of solid angle and 1rad means 1 radian radian is the SI unit of plane angle. Plane angle is defined as:-
An angle of one radian results in an arc with a length equal to the radius of the circle.
Solid angle is defined as:-
the solid angle subtented at the centre of a sphere of radius r by a portion of the sphere whose area A,equals r square

 

[Priya123]

and this is physics not maths it is under the topic of measurements

 

[Ken G]

In case it isn't clear from the above, 1 steradian is the solid angle of 1/4pi fraction of the full field of view, the full sphere. So the physical area of an object at distance D that is curved in the shape of a piece of a spherical shell, and subtends 1 steradian of solid angle, is D², and an object at that same distance that has a smaller than 1 steradian solid angle is proportionately smaller in area. (Usually we imagine it is not in the shape of a piece of a spherical shell, but it is also much much smaller than 1 steradian, so the area here is just the cross sectional area of the object.)

The radian works the same but in length-- an object that subtends 1 radian at distance D has length D along a piece of a circular arc. Again the formula becomes more useful for objects much smaller than 1 radian in angle, so the length is a kind of length perpendicular to the line of sight, rather than literally worrying about it being shaped like the arc of a circle.

 

[Priya123]

thats a more clearer way of my explanation and thanks it helped me out too

 

[jtbell]

The original question was asked two and a half years ago, by the way.