well saying that the radiation from the sun all travels at the same speed

, this radiation will cover a spherical shape of area.(just like blowing a balloon)

now the distance of the earth to the sun is about

this means you need to find the surface area of a sphere with a radius of , then you need to find the area of intersection between the sphere and the earth(assume to be in the middle of the earth)

surface area of the sphere:

where is the radius of the earth and has the value of

hence the area of intersection is:

where is the radius of the earth and has the value of . The area of intersection is can be assumed as a circle since the distance of the earth from the sun is very much larger tahn the radius of the earth.

hence the fraction of radiation received by earth is