Question

Which explanation can be used to derive the formula for the circumference of a circle?

Find the relationship between the circumference and the diameter by dividing the length of the circumference and length of the diameter. Use this quotient to set up an equation to showing the ratio of the circumference over the diameter equals to π . Then rearrange the equation to solve for the circumference. Substitute 2 times the radius for the diameter.

Find the length of the radius and the length of the diameter. Then, write a ratio comparing the length of the radius to the diameter. Multiply the ratio by ​ π ​ and set it equal to the circumference.

Find the length of the diameter. Square this length and write a ratio for the square of the diameter to the radius. Set the ratio equal to the circumference.

Find the length of the diameter. Square this length and write a ratio for the square of the diameter to the radius. Set the ratio equal to the circumference. Find the relationship between the area and the radius. Then set up an equation showing the ratio of the area to the radius. Substitute the area for 3 times the circumference. Finally, rearrange the equation to solve for the circumference.

Answer 1

Find the length of the diameter and the length of the circumference of the circle. Divide the length of the circumference by the length of the diameter. Set up an equation showing the ratio of the circumference to the diameter equal to ​ π ​. Then rearrange the equation by solving it for the circumference. Substitute 2 times the radius for the diameter.

Answer 2

Answer: The correct answer is (A) Find the relationship between the circumference and the diameter by dividing the length of the circumference and length of the diameter. Use this quotient to set up an equation to showing the ratio of the circumference over the diameter equals to π . Then rearrange the equation to solve for the circumference. Substitute 2 times the radius for the diameter.

Step-by-step explanation: We are given to select the correct explanation among the options about the derivation of the formula for the circumference of the circle.

We know that the ratio of the circumference to the diameter is equal to the irrational number pi.

That is, if 'C' be the circumference and 'd' be the diameter of a circle, then

`(C)/(d)=\pi.`

Now, since diameter is twice the radius, so

`d=2r,~\textup{where 'r' is the radius}.`

Therefore, the circumference 'C' is given by

`C=2\pi r.`

Thus, the correct explanation is the first one.