Understanding the Relationship between Circles and the number Pi
 Understanding the Relationship between Circles and the Number Pi
 Parts of a Circle
 Circumference of a Circle
 Calculating the Circumference – StepbyStep Instruction
 Understanding the Relationship between Circles and the number Pi – Summary
 Relationship between Circles and the number Pi – Frequently Asked Questions
Basics on the topic Understanding the Relationship between Circles and the number Pi
Understanding the Relationship between Circles and the Number Pi
Pi, represented by the Greek letter $\pi$, is a fundamental constant in mathematics that appears in many formulas relating to circles and other aspects of geometry and physics. It is the ratio of a circle's circumference (the distance around the circle) to its diameter (the distance across the circle through its center).
The Number Pi
Pi ($\pi$) is a special number in mathematics that represents the ratio of a circle's circumference ($C$) to its diameter ($d$). This means that no matter how big or small a circle is if you divide the circumference (the distance around the circle) by the diameter (the distance across the middle of the circle), you will always get approximately the same number: pi, or π.
A common approximation for pi is $\frac{22}{7}$, which is close but not exactly $\pi$. Pi is actually an infinite, nonrepeating decimal, meaning it has no exact value and no final digit. This type of number is known as an irrational number.
Parts of a Circle
A circle is a perfect round shape, where every point on the edge is the same distance from the center. This consistent distance is known as the radius, and it is half the length of the diameter. For more details on the fundamental characteristics of a circle, see what is a circle.
Term  Definition 

Radius ($r$)  The distance from the center of the circle to any point on its edge. 
Diameter ($d$)  The distance across the circle, passing through the center. It is twice the radius. 
Circumference ($C$)  The distance around the circle. It is calculated as $\pi$ times the diameter of the circle ($\pi d$). 
The radius of a circle is half the diameter because it spans from the center of the circle to its edge, effectively splitting the diameter into two equal parts. Thus, the radius is always half the length of the diameter.
Circumference of a Circle
The circumference of a circle can be calculated using the formula: $C = \pi d$, where $C$ stands for circumference and $d$ is the diameter of the circle.
The circumference of a circle can also be calculated using $C = 2\pi r$, where $C$ is the circumference and $r$ is the radius. This formula is derived from $C = \pi d$ by substituting $d$ with $2r$ since the diameter is always twice the radius.
Calculating the Circumference – StepbyStep Instruction
Example 1 Find the circumference of a circle with a diameter of 8 inches.
 Identify the formula for circumference when you know the diameter: $C = \pi d$.
 Substitute the given diameter into the formula: $C = 3.14 \times 8$ inches.
 Calculate the product to find the circumference: $C = 25.12$ inches.
Example 2 Calculate the circumference of a circle with a radius of 5 meters.
 Use the formula for circumference with radius: $C = 2\pi r$.
 Substitute the radius into the formula: $C = 2 \times 3.14 \times 5$ meters.
Complete the multiplication to find the circumference: $C = 31.4$ meters.
Apply the circumference formula using diameter: $C = \pi d$.
 Plug in the diameter: $C = 3.14 \times 14$ centimeters.
 Calculate to find the circumference: $C = 43.96$ centimeters, which rounds up to approximately 44 centimeters.
Try some on your own.
Understanding the Relationship between Circles and the number Pi – Summary
Key Learnings from this Text:
Pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter.
The circumference of a circle can be calculated using the diameter with the formula $C = \pi d$ or using the radius with $C = 2\pi r$.
Essential circle terms include radius, diameter, and circumference, defining the basic properties of circular shapes.
The text includes illustrations and practical examples to demonstrate circumference calculations and the relevance of Pi in realworld scenarios.
Understanding Pi provides a foundation for more complex geometric and mathematical studies.
Relationship between Circles and the number Pi – Frequently Asked Questions
Transcript Understanding the Relationship between Circles and the number Pi
As he does every morning, Farmer Johnson is enjoying a hot cup of joe before he gets to his morning chores. But in the distance, he sees something strange. Sakes alive! What's that? Crop circles everywhere! Only one kind of varmint could leave those tracks aliens! Farmer Johnson takes careful measurements of the mysterious markings and hightails it back to the homestead to get to the bottom of it! While making sense of the crop circles and trying to prove that aliens exist, Farmer Johnson will learn about "Circles and the Number Pi". Farmer Johnson first examines the biggest circle, looking at the distance from the center to different points on the edge of the circle. What do you notice about these measurements? Every time Farmer Johnson draws another line, he gets the same length! 420 feet. Well I'll be a monkey's uncle! Is this proof the markings had to have been made by aliens? Well, actually no, Farmer Johnson. This pattern is true for all circles, no matter who makes them. We call this measurement, from the center of a circle to any point on its edge, the radius. It's usually represented by the letter 'r'. No matter which point on the edge you pick, the radius will always be the same length. In fact, that's what makes a circle a circle! It's a shape formed by all the points that are the same distance away from the center. Farmer Johnson studies his board carefully. There's got to be some way to prove these circles are the work of alien invaders! This time, Farmer Johnson measured both the distance from the center of the circle to the edge as well as the distance across the crop circles, going through the center of the circle. Do you notice any relationship between these lengths? What the blazes! This length across the circle is exactly twice the radius! Here, 840 feet is equal to 2 times 420 feet. And it's true for the other circles too! Farmer Johnson thinks this can't be no piddlin' coincidence! Is it hard and fast proof that aliens were behind the circles? Nope, Farmer Johnson. It's just math. This line is called the diameter. It passes through the center of a circle and touches opposite points on the circle's edge and is always equal to twice the radius. Why do you think that's so? A diameter is made by putting two radii together. Farmer Johnson is still fixin' to prove his wild notion about aliens. He just needs a big breakthrough, something that mere mathematics can't explain! This time, Farmer Johnson measures the distance around the circles, tracing the entire edge. We call this measurement the circumference. Hmm, Farmer Johnson reckons he spies a relationship between the circumference and the diameter. The bigger the circle is across, the bigger it is around. Let's examine the relationship more closely by calculating the ratio of the circumference to the diameter. Two thousand, six hundred and thirty eight divided by 840 is equal to approximately 3.14. Jiminy Cricket! Do you see what Farmer Johnson sees? The ratio, rounded to the nearest hundredth, is always the same! This can only be the work of those blasted aliens! Now, hold your horses, Farmer Johnson. It looks like Miz Johnson has a thing or two to teach her husband. This 'mysterious' ratio you were hollering about is actually a number that mathematicians have known for centuries. We call it PI, and here's the symbol we use to represent it. It's the ratio of any circle's circumference to it's diameter. You heard right, it doesn't matter what size circle you have, the ratio is always the same. Here's the value of pi, 3.141519265 and so on. The number pi is a nonterminating decimal, which means it never ends. It also doesn't repeat. That means it's an irrational number. If we need to calculate with this number, we can estimate it using either the decimal form 3.14 or the fraction 22 over 7. If we need to be exact, we can leave the symbol PI in our answer. Isolating the variable 'c' from, pi equals 'c' over 'd', gives us the formula for the circumference of a circle. Circumference equals pi times diameter. Because the diameter of a circle is twice its radius... we can also write this formula as circumference equals 2 times pi times radius. Farmer Johnson's all tore up that aliens didn't make the crop circles afterall but in the meantime, let's review. All circles share some basic properties, no matter their size. The radius, which is a line drawn from the center of the circle to the outer edge...is always the same length, no matter how you draw it. The diameter is any line that crosses through the center of the circle and ends at two opposite edges. And it's always twice the size of the radius. Finally, we have the circumference of a circle. That's the distance around the circle. The ratio of the circumference to the diameter is the number PI, written with the symbol pi. In calculations, pi is often estimated as 3.14 or 22 over 7 and it definitely wasn't created by aliens! Farmer Johnson heads back to his water tower, down in the mouth that aliens don't exist afterall.
Understanding the Relationship between Circles and the number Pi exercise

Identify measurements of a circle.
HintsThe 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.
SolutionThe 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.
HintsPi 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 $\sqrt{2}$.
SolutionPi 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.
HintsThe 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 $5$.
SolutionThe 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.
HintsThe $r$ 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.
SolutionThe 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.
SolutionThe 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.
HintsThe 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 $\pi$.
SolutionTo 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