Question

The circumference of Circle K is pi. The circumference of Circle L is 4xpi.

Two circles, one labeled “Circle K” and the other as “Circle L.”

What is the value of the ratio of their circumferences? of their radii? of their areas? Write the ratios as fractions in the simplest form.

Answer 1

Ratio of circumferences:
`\displaystyle(1)/(4)`

Ratio of radii:
`\displaystyle(1)/(4)`

Ratio of areas:
`\displaystyle(1)/(16)`

Explanation:

Hi there!

We are given:

- The circumference of Circle K is
`\pi`

- The circumference of Circle L is
`4\pi`

Therefore, the ratio of their circumferences would be:

`\displaystyle(\pi)/(4\pi)` ⇒
`\displaystyle(1)/(4)` when simplified

The formula for circumference is
`C=2\pi r`, where r is the radius. To find the ratio of the circles' radii, we must identify their radii through their given circumferences.

If the circumference of Circle K is
`\pi`, or
`1\pi`, then its radius is
`\displaystyle(1)/(2)`.

If the circumference of Circle L is
`4\pi`, then its radius is
`\displaystyle(4)/(2)`, which is 2.

Therefore the ratio their radii would be:

`\displaystyle\frac{(1)/(2)}{{2}}` ⇒
`\displaystyle(1)/(2)*(1)/(2)` ⇒
`\displaystyle(1)/(4)` when simplified

The formula for area is:

`A=\pi r^2`

First, let's find the area of Circle K:

`A=\pi (\displaystyle(1)/(2))^2\\\\A=\displaystyle(1)/(4)\pi`

Now, let's find the area of Circle L:

`A=\pi (2)^2\\A = 4\pi`

Therefore, the ratio of their areas would be:

`\displaystyle((1)/(4)\pi)/(4\pi)` ⇒
`\displaystyle((1)/(4))/(4)` ⇒
`\displaystyle(1)/(4) * (1)/(4)` ⇒
`\displaystyle(1)/(16)` when simplified

I hope this helps!