# Understanding the Relationship between Circles and the number Pi

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## Basics on the topicUnderstanding the Relationship between Circles and the number Pi

After this lesson, you will be able to identify and define parts of a circle, plus explain the where the number pi comes from.

The lesson begins by teaching you the parts of a circle: radius, diameter, and circumference. It leads you to examine the ratio between a circle’s diameter and circumference, which is pi. It concludes with a definition of pi, which is often approximated by 22/7 or 3.14.

Learn about circles and the number pi by helping Farmer Johnson prove that aliens exist!

This video includes key concepts and vocabulary such as circle (a collection of points equidistant from the center); radius (a line from a circle’s center to edge), diameter (a line passing through a circle’s center, connecting two points on the edge), circumference (the distance around a circle), and pi (the ratio between a circle’s circumference and diameter).

Before watching this video, you should already be familiar with ratios and how to represent values as both fractions and decimals. You should also be comfortable with the concept of a circle.

After watching this video, you will be prepared to learn how to apply the number pi in other formulas, like the area of a circle.

Common Core Standard(s) in focus: 7.G.B.4, 7.G.B.6 A video intended for math students in the 7th grade Recommended for students who are 12-13 years old

## Understanding the Relationship between Circles and the number Pi exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Understanding the Relationship between Circles and the number Pi.
• ### Identify measurements of a circle.

Hints

The radius of a circle measures the distance from the center to any part on the edge of the circle.

The diameter is the distance through the center of the circle from one edge to the other.

The circumference is the distance around the outside edge of a circle.

Solution

The circumference is $942.5$ ft and refers to the distance around the outside of the circle.

The diameter is $300$ ft and this measurement is the distance across the circle, passing through the center.

The radius is half of the diameter and is $150$ ft. This measurement is the distance from the center of the circle to the outside.

• ### Understand the value of Pi.

Hints

Pi is a special number that mathematicians use when they are talking about circles. It's the number you get if you divide the distance around a circle (the circumference) by the distance across the middle of the circle (the diameter).

Rational numbers are numbers that can be written as a fraction, where both the top number (numerator) and the bottom number (denominator) are whole numbers. For example, $\frac{1}{2}$, $3$, and $\frac{-4}{5}$ are all rational numbers.

Irrational Numbers: Irrational numbers are numbers that can't be written as a simple fraction. Their decimal places go on forever without repeating any pattern. Examples of irrational numbers include $\pi$ and the $\sqrt{2}$.

Solution

Pi is a number that mathematicians have known for centuries and the symbol is this: $\bf{\pi}$. The ratio of a circle's circumference to its diameter is the value of Pi. The value of Pi is approximately $\bf{3.14...}$ and is known as an irrational number.

• ### Determine the ratio equivalent to the value of Pi.

Hints

The ratio of the value of $\pi$ is $\dfrac{C}{d}$.

To find the value of a ratio, division can be used.

For example, the ratio of $\dfrac{15}{3}$ can be simplified to $\dfrac{5}{1}$, or $5$.

Solution

The ratio $\dfrac{22}{7}$ simplifies to $\pi$. This can be found by dividing $22$ by $7$, to get approximately $3.14...$.

• ### Using a formula to find the circumference of a circle.

Hints

The $r$ in the formula stands for the radius, which is the distance from the center of the circle to the outside.

The $d$ in the formula stands for the diameter, which is the distance across the circle passing through the center.

To use a formula, substitute in values you know into the appropriate variable.

For example, if we knew we had a diameter of $9$ inches, and we were using the formula $C=\pi d$ to find the circumference, we would replace the $d$ with the $9$ like this: $C=\pi (9)$.

The radius is half the length of the diameter.

Solution

The circle has a diameter of $12$ cm, therefore the radius is $6$ cm.

The two equations that can be used to find the circumference are:

$C=\pi(12)$

$C=2 \pi (6)$

• ### Identify the ratio of Pi.

Hints

$C=\text{circumference}$

$d=\text{diameter}$

$r=\text{radius}$

Ratios must be written in a specific order in order for them to be accurate.

The ratio $\frac{1}{2}$ has a different value than $\frac{2}{1}$.

The measurements of a circle are depicted here, as well as the ratio of Pi.

Solution

The ratio for Pi is $\pi=\dfrac{C}{d}$.

If the circumference is divided by the diameter the value is always approximately $3.14$.

• ### Determine the circumference of a circle.

Hints

The formula to find the circumference of a circle is $C=\pi d$, or $C=2 \pi r$.

The radius is half the distance of the diameter.

To round a number to the nearest tenth, look at the number in the hundredth place. If it's 5 or more, round the tenth's place up; if it's less than 5, keep the tenth's place the same.

A calculator will come in handy to help you solve using the value of $\pi$.

Solution

To find the circumference of the circle shown, you can choose to use the given radius of $4$ cm, or double it for the diameter of $8$ cm.

Here are both ways shown with the two different formulas.

$C=\pi d$

$C=\pi(8)$

$C=25.1$ cm

${}$

$C=2 \pi r$

$C=2 \pi (4)$

$C=25.1$ cm