Plane angle VS Solid angle

• Plane angle

It refers to an angle in 2 dimensional figure like square, triangle, circle etc.

 

There are defined by two types.

 

1. a plane angle defined by 60 minutes

It is an angle notation that is usually used by ordinary people.

As shown below, the 60minute method separates the circle into 360 pieces, which is 6times of 60, and then the angle of one piece defines as 1∘.

 

And  1∘ divided by 60 minutes, 1 minute is divided by 60 seconds.

The reason why it called 'angle defined by 60 minutes' is that 1∘ is 60 minutes.

 

- right angles

 

 

- acute angles : a - obtuse angles : b - flat angles : c

 

2. a plane angle defined by the arc method

The angle method using the length of the arc is called the arc method.

The circumference ratio is the ratio between the circumference length(C) and the diameter(d) as shown below.

 

- Formula(the ratio of the circumference of a circle to its diameter)

But we use the radius of the circle more than the diameter, so If you use the radius r, it's like below.

 

- Formula(θ)

Then any angle θ can be determined by the length of the arc(s), and the radius(r). If s=r the, it means 1 radian.

 

- Formula(s)

r=s=1 rad r=2s=2 rad r=3s=3 rad ... the angle of the half lap=π rad=3.14 rad s=rθ

 

• Solid angle

It refers to an angle in 3 dimensional figure like cube, cuboid, sphere etc.
In three dimensions what a surface subtends on a point with the help of surfaces like cones and straight lines is a solid angle.

- Formula(total surface S)

 

reference : https://ballpen.blog/%ea%b0%81%eb%8f%84-%ea%b0%9c%eb%85%90-%ec%9d%b4%ed%95%b4/