In a circle with a radius of 5 inches, what is the circumference? Use the formula C = 2pr.
The correct answer and explanation is :
To find the circumference of a circle, we can use the formula:
[
C = 2\pi r
]
where:
- (C) represents the circumference,
- (\pi) (pi) is approximately 3.1416,
- (r) is the radius of the circle.
Step-by-Step Solution:
In this case, the radius of the circle is given as 5 inches. Therefore, we can substitute (r = 5) inches into the formula.
[
C = 2 \times \pi \times 5
]
Now, substitute the value of (\pi) (approximately 3.1416):
[
C = 2 \times 3.1416 \times 5
]
First, multiply 2 and 3.1416:
[
2 \times 3.1416 = 6.2832
]
Next, multiply this result by 5:
[
6.2832 \times 5 = 31.416
]
Thus, the circumference of the circle is approximately 31.416 inches.
Explanation:
The formula (C = 2\pi r) is derived from the fact that the circumference of a circle is the distance around the circle, which is proportional to its radius. The constant (\pi) is a mathematical constant that represents the ratio of a circle’s circumference to its diameter. This ratio is the same for every circle, and its value is approximately 3.1416.
To understand why the formula works, consider that the circumference is directly proportional to the radius. If the radius were doubled, the circumference would also double. The factor of 2 in the formula accounts for the fact that the radius is half of the diameter (the full width of the circle across the center), and the formula involves multiplying by the entire diameter.
In this problem, we were asked to calculate the circumference for a circle with a radius of 5 inches, and after applying the formula, we determined that the circumference is approximately 31.416 inches. This measurement tells us the length of the boundary around the circle.