What is difference between plane angle and solid angle?

A plane angle is one that can be measured completely within a plane. This may be because the angle occurs in a 2-D space, or it is an angle that is contained completely within a plane that is in a 3-D (or higher-D) space.

So the angles in a triangle are plane angles, because they are in a plane. Similarly, the angles in a spherical triangle are plane angles, because even though they measure things in a 3-D space, they can always be shown to be wholly within a plane.

Another way to consider a plane angle is to think of angles within a circle. You can divide the circle up into 2π radians, for example, or 360°, or 400 gon, or whatever units you choose. However, the radian as a unit is defined by the subtended arc length that is equal to the radius of the circle.

A steradian, the basic unit, is solid angle is one that is subtended by a portion of the surface area of a sphere that is equal to r^2, where r is the radius of the sphere. It can be considered to be a cone subtended at the center of the sphere by a part of the surface of the sphere.

A sphere has a surface area of 4πr^2, so a sphere contains 4π steradians.

Credit: https://www.quora.com/What-is-difference-between-plane-angle-and-solid-angle