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 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 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 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.

Step-by-step explanation: We are given five different options and we are to select which explanation is correct to derive the formula for a circumference of a circle.

Let 'C' be the circumference and 'd' be the diameter of a circle. Now, we will write the ratio of the circumference to the diameter as

[tex]\dfrac{\textup{C}}{\textup{d}}.[/tex]

Also, we know that

[tex]\dfrac{\textup{C}}{\textup{d}}=\pi.[/tex]

And diameter of a circle is twice the radius, so

[tex]d=2r.[/tex]

Therefore,

[tex]\dfrac{\textup{C}}{2\textup{r}}=\pi\\\\\Rightarrow \textup{C}=2\pi \textup{r}.[/tex]

This is the formula for the circumference of a circle. Since this explanation matches exactly with the first option, so the correct option 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.