# Question #a078d

Dec 14, 2017

see explanation

#### Explanation:

A: circumference formula: $c = 2 \pi r = \pi d$
therefore, $\pi = \frac{c}{d}$
so, for circle A, you get: $\frac{21.98}{7} = 3.14$
for circle B, you get: $\frac{18.84}{6} = 3.14$

B: area formula: $a = \pi {r}^{2}$, where $r = \frac{d}{2}$
therefore, $\pi = \frac{a}{r} ^ 2$
for circle A, you get: $\frac{38.465}{{3.5}^{2}} = 3.14$
for circle B, you get: $\frac{28.26}{{3}^{2}} = 3.14$

C: You observe that the value of pi is the same in all cases. The value of pi is a CONSTANT.

GOOD LUCK

Dec 14, 2017

$\pi = 3.14$

#### Explanation:

$C = \pi D = 2 \pi r$, where C is the circumference, D is the diameter, and r is the radius.
$A = \pi {r}^{2}$ where A is the area and r is the radius.

a)

We need to find a value of $\pi$ using the circumference formula.

$C = \pi D$
$\pi = \frac{C}{D}$

For Circle A:

$\pi = \frac{21.98}{7}$
$\pi = 3.14$

For Circle B:

$\pi = \frac{18.84}{6}$
$\pi = 3.14$

b)

$A = \pi {r}^{2}$

We aren't given the radius of the circle in each case, but the diameter is double the radius, $D = 2 r$ or $r = \frac{D}{2}$.
Substituting into our area formula we get:

$A = \pi {\left(\frac{D}{2}\right)}^{2}$
$A = \pi {D}^{2} / 4$
$A = \frac{1}{4} \pi {D}^{2}$

Rearranging for $\pi$:

$4 A = \pi {D}^{2}$
$\pi = \frac{4 A}{D} ^ 2$

For circle A:

$\pi = \frac{4 \times 38.465}{{7}^{2}}$
$\pi = 3.14$

For circle B:

$\pi = \frac{4 \times 28.26}{6} ^ 2$
And no prizes for guessing that this evaluates to:

$\pi = \frac{3}{14}$

So what do we notice?

This is going to go into more detail than you need for your homework, although you may wish to read it; I personally find this stuff quite interesting.

$\pi = 3.14$
$\pi = 3.14$
$\pi = 3.14$
$\pi = 3.14$

In fact, $\pi$ takes the same value in all contexts. $\pi$ is a constant, which is approximately equal to

$\pi \approx 3.141592653589793238462643383279502884 \ldots$

No wonder we often round it to $3.14$

$\pi$ is itself defined as the ratio of a circle's diameter to its radius. It has no exact value; the decimal of pi goes on forever. This is different however to the infite decimal of $\frac{1}{3}$.

$\frac{1}{3} = 0.333333333333333 \ldots = 0. \dot{3}$

This decimal will repeat. So too will any fraction you can come up with. $\pi$ is different; it never reccurs, and it cannot be written as a fraction. Since it cannot be written as a fraction, $\pi$ is an irrational number. Other irrational numbers include irrational square roots, such as $\sqrt{2}$. $\pi$ is different to other irrational numbers, however. With $\sqrt{2}$, we can use the hypoteneuse of a $\text{1-1-} \sqrt{2}$-right-triangle to plot the value of this number on a number line. We cannot do anything like this with $\pi$, so as well as being irrational, pi is a transcendental number.

Both of these mean we can never actually describe pi accurately as a decimal. In fact, if we wanted to be accurate, we would leave the answers in terms on $\pi$. Circle A has a circumference of exactly $7 \pi$ inches (who uses inches in this day and age anyway? regardless) and an area of $\frac{49}{4} \pi$ square inches. I mean, we could use pi to a million decimal places if we wanted, but this wouldnt be as accurate as leaving an answer it terms of pi. The same applies to how you leave answers as square roots.