5.3 Decimals and fractions  (Page 3/8)

Archimedes discovered that for circles of all different sizes, dividing the circumference by the diameter always gives the same number. The value of this number is pi, symbolized by Greek letter $\pi$ (pronounced pie). However, the exact value of $\pi$ cannot be calculated since the decimal never ends or repeats (we will learn more about numbers like this in The Properties of Real Numbers .)

If we want the exact circumference or area of a circle, we leave the symbol $\pi$ in the answer. We can get an approximate answer by substituting $3.14$ as the value of $\text{π}.$ We use the symbol $\approx$ to show that the result is approximate, not exact.

Properties of circles

$\begin{array}{c}r\text{is the length of the radius.}\hfill \\ d\text{is the length of the diameter.}\hfill \end{array}$

$\begin{array}{cccc}\text{The circumference is}2\text{π}\mathit{\text{r}}.\hfill & & & C=2\text{π}\mathit{\text{r}}\hfill \\ \text{The area is}\text{π}{\mathit{\text{r}}}^2.\hfill & & & A=\text{π}{\mathit{\text{r}}}^2\hfill \end{array}$

Since the diameter is twice the radius, another way to find the circumference is to use the formula $C=\text{π}\mathit{\text{d}}.$

Suppose we want to find the exact area of a circle of radius $10$ inches. To calculate the area, we would evaluate the formula for the area when $r=10$ inches and leave the answer in terms of $\text{π.}$

We write $\pi$ after the $100.$ So the exact value of the area is $A=100\text{π}$ square inches.

To approximate the area, we would substitute $\pi \approx 3.14.$

Remember to use square units, such as square inches, when you calculate the area.

Approximate $\pi$ With a fraction

Convert the fraction $\frac{22}{7}$ to a decimal. If you use your calculator, the decimal number will fill up the display and show $3.14285714.$ But if we round that number to two decimal places, we get $3.14,$ the decimal approximation of $\text{π}.$ When we have a circle with radius given as a fraction, we can substitute $\frac{22}{7}$ for $\pi$ instead of $3.14.$ And, since $\frac{22}{7}$ is also an approximation of $\text{π},$ we will use the $\approx$ symbol to show we have an approximate value.