The circumference of a circle is 28 pi inches, what is the length of the radius of this circle?
The correct answer and explanation is:
Answer:
The radius of the circle is 14 inches.
Explanation:
The circumference CC of a circle is related to its radius rr by the formula: C=2πrC = 2 \pi r
Given that the circumference is 28π28 \pi inches, we can set up the equation: 28π=2πr28 \pi = 2 \pi r
To solve for rr, first divide both sides by π\pi: 28=2r28 = 2r
Next, divide both sides by 2: r=282=14r = \frac{28}{2} = 14
Thus, the radius rr is 14 inches.
Detailed Explanation:
The circumference is the distance around the edge of the circle. It depends directly on the radius, which is the distance from the center of the circle to any point on the edge.
The formula C=2πrC = 2 \pi r is derived from the definition of π\pi, which is the ratio of the circumference to the diameter (d=2rd = 2r) of any circle: π=Cd ⟹ C=πd=π(2r)=2πr\pi = \frac{C}{d} \implies C = \pi d = \pi (2r) = 2 \pi r
When you are given the circumference, you can rearrange this formula to find the radius: r=C2πr = \frac{C}{2\pi}
In this problem, the circumference is expressed as 28π28 \pi, which already includes π\pi. Dividing by 2π2 \pi effectively cancels the π\pi and divides 28 by 2, leaving you with the radius.
This kind of problem is common in geometry and helps reinforce understanding of the fundamental properties of circles. Knowing how to rearrange and solve for variables in formulas is a crucial algebra skill. Here, the key was to isolate the radius variable and simplify correctly.
The radius of 14 inches means that the distance from the center of this circle to any point on its edge is 14 inches, which also implies the diameter (twice the radius) is 28 inches.