Are "within a radius of 10 meter" and "in a radius of 10 meter" synonymous? I was told that "within a radius" meant everyone within the circle, and "in the radius" meant everyone on the border of the circle, but is it true? Most people use "in a radius" to mean "within a radius".
These are nearly synonymous phrases. Within is being used as a preposition to indicate that you are referencing the area inside of the radius in relation to the radius. In basically communicates the same message, but with less focus on the actual radius.
However, “within” and “in” are not always interchangeable. “Within” describes an object’s placement for location and time.
- The building is within the city limits
- You will receive your order within five business days
“Within” is specifically used to describe the boundaries of a location or time, consequently placing a larger emphasis on constraints. “Within six hours” implies that something will happen before six hours. However, “in six hours” suggests a more flexible time range (something will happen in about six hours).
“In” is used for a wider variety of situations, such as inclusion (there are many more), and is used in many idiomatic expressions as well.
- She serves in the military
- The delivery came just in time
Additionally, the word “meter” in your phrases should be plural because multiple meters are modifying the radius.
To answer the second part of your question, I believe “on the border/edge/circumference of the circle” most clearly communicates that you are referencing the border of the circle and not the area it encloses.
Another note: you can rewrite the phrases as “within/in a 10 meter radius” to make them more concise. This has no effect on the meaning.
If you are speaking mathematically, the distinctions that you are looking for can be expressed unamiguously
less than 10 meters from the circle's center
10 meters from the circle's center
no more than 10 meters from the circle's center.
Informally, "within 10 meters from X" literally excludes being exactly 10 meters from X, but in actual usage it frequently means "no more than 10 meters from X." If you want to be unambiguous, I recommend saying "less than 10 meters from X" rather "within 10 meters of X."
The phrase "in the radius" to mean "on the circumference" is mathematically nonsensensical, and I have never heard it even in very informal speech.