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 – Step-by-Step Instruction
- Understanding the Relationship between Circles and the number Pi – Summary
- Relationship between Circles and the number Pi – Frequently Asked Questions
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Learning text 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 centre).
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 $\frac{22}{7}$, or 3.141592 to six decimal places, which is close but not exactly $\pi$. Pi is actually an infinite, non-repeating 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 centre. 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 centre of the circle to any point on its edge. |
| Diameter ($d$) | The distance across the circle, passing through the centre. 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 centre 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 – Step-by-Step 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 to 2 decimal places.
Example 2 Calculate the circumference of a circle with a radius of 5 metres.
- Use the formula for circumference with radius: $C = 2\pi r$.
- Substitute the radius into the formula: $C = 2 \times 3.14 \times 5$ metres.
- Complete the multiplication to find the circumference: $C = 31.4$ to 1 decimal place metres.
Example 3 Determine the circumference of a circle where the diameter is 14 centimetres.
- Apply the circumference formula using diameter: $C = \pi d$.
- Plug in the diameter: $C = 3.14 \times 14$ centimetres.
- Calculate to find the circumference: $C = 43.96$ centimetres, which rounds up to approximately 44 centimetres.
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 real-world scenarios.
- Understanding Pi provides a foundation for more complex geometric and mathematical studies.
Relationship between Circles and the number Pi – Frequently Asked Questions
Understanding the Relationship between Circles and the number Pi exercise
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Identify measurements of a circle.
HintsThe radius of a circle measures the distance from the centre to any part on the edge of the circle.
The diameter is the distance through the centre 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 centre.
The radius is half of the diameter and is $150$ ft. This measurement is the distance from the centre of the circle to the outside.
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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.
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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 $3.14...$.
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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 centre 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 centre.
To use a formula, substitute in values you know into the appropriate variable.
For example, if we knew we had a diameter of $9$ cm, 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:
$C=\pi(12)$
$C=2 \pi (6)$
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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 $3.14$.
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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
