Final answer:
The radius of a circle with circumference 3πa/4 can be found by using the formula C=2πr and solving for r, resulting in r=3a/8. This solution aligns with the geometric relationship where the diameter of the circle fits neatly into a square with the side length 'a'.
Step-by-step explanation:
If a circle has a circumference of 3πa/4, where 'a' represents a unit length and is non-zero, we can find the radius of the circle by using the formula for the circumference of a circle, which is C=2πr (where C is the circumference and r is the radius). In this instance, the circumference is given as 3πa/4.
To solve for 'r', we equate the given circumference to the formula:
Divide both sides by π to simplify:
- 3a/4 = 2rNow we solve for 'r' by dividing both sides of the equation by 2:
Since the task is to fit the circle inside a square such that a = 2r, we can check our result by doubling the radius to find the diameter, which should equal 'a'. Indeed, if r = 3a/8, then the diameter is 2r = 3a/4, thus satisfying the condition.
This method of determining the radius is based on principles that were already used by the ancient Greeks and remain an essential part of geometry.