How to find the ratio of diameter and circumference - ACT Math

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Question

Let represent the area of a circle and represent its circumference. Which of the following equations expresses in terms of ?

Answer

The formula for the area of a circle is $A=\pi r^2$ , and the formula for circumference is $C=2\pi r$ . If we solve for C in terms of r, we get
$r=C/2\pi$ .

We can then substitute this value of r into the formula for the area:

$A=\pi r^2$

$=\pi (C/2\pi )^2$

$=C^2\pi /4\pi ^2$

$=C^2/4\pi$

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Question

A circle has a radius of . What is the ratio of its circumference to its area?

Answer

1. Find the circumference:

$C=\pi d=2\pi r$

$C=2\cdot \pi \cdot 2=4\pi$

2. Find the area:

$A=\pi r^{2}$

$A=\pi \cdot (2^{2})=4\pi$

3. Divide the circumference by the area:

$\frac{C}{A}=\frac{4\pi }{4\pi }=1$

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Question

How far must a racehorse run in a lap race where the course is a circle with an area of $36\pi$ ?

Answer

To find the answer, we need to first find the circumference of the circle then mulitply that by 4 because the racehorse is running 4 laps.

We know that the area of the circle is $36\pi$ . We can then find the radius.

$A=\pi r^2$

$36\pi=\pi r^2$

Because , our diameter is 12.

We can now find the circumference.

$C=d\pi$

$C=12 \pi$

Multiply the circumference by 4 (our racehorse is running 4 laps) to find the answer.

$12 \pi \cdot 4=48\pi$

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