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Here’s the relationship between the circumference and diameter of a circle in three key points:

1. Formula: The circumference C of a circle is directly proportional to its diameter D. The relationship is given by the formula:

C=πD

where π (pi) is approximately 3.14159.

2. Proportionality: For any circle, the ratio of the circumference to the diameter is constant and equal to π. This means the circumference is roughly 3.14 times the diameter.

3. Scalability: If the diameter of a circle doubles, the circumference also doubles, maintaining the same proportional relationship.

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Bingo Task Activities Instructions Consolidated

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Bingo Task Activities Instructions Consolidated

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1. When submitting, always remember to include your Group Name.

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Joel

Kayden

Danzel

Yisen

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1. Celestine

2. Adele

3. Brilynn

4. Casey

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Relationship Between Circumference & Diameter

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Change the circumference and diameter of a circle to see the relationship.

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Watch from 4'50 - 5'55

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Group 1

1. Fundamental Geometric Relationship and the Role of Pi (π) in Defining Circularity: The relationship between the diameter (d) and circumference (C) of a circle is elegantly encapsulated in the formula C=π×dC=π×d, where π (pi) is the mathematical constant approximately equal to 3.14159. This formula reflects an intrinsic property of all circles, regardless of their size. Pi, an irrational number, serves as the scaling factor between the diameter and circumference of any circle. Geometrically, this relationship comes from the way a circle is defined: it is the set of all points that are equidistant from a central point, with the distance from the center to any point on the circle being the radius. The diameter is simply twice the radius, and thus, the circumference, or the perimeter of the circle, is always a little more than three times its diameter. Pi’s role in this formula is crucial as it reflects the constant ratio between these two key measurements. The fact that π is irrational means that it cannot be expressed as a simple fraction, and its decimal expansion never ends or repeats, which makes the connection between the diameter and circumference even more fascinating and unique to circles. This relationship transcends simple measurements and enters the realm of deeper mathematical significance, showing that despite the infinite nature of pi, the formula C=π×dC=π×d always holds true. This geometric constant is not only fundamental to the properties of a circle but is also central to many other branches of mathematics and science, particularly in fields involving periodicity, waveforms, and rotational motion.

2. The Proportionality of Diameter and Circumference Across All Circles: One of the most striking aspects of the relationship between the diameter and the circumference is the proportionality between the two measurements. The circle is a shape of constant proportionality, meaning that no matter how large or small the circle is, its circumference will always be about 3.14159 times its diameter. This proportionality is not just an interesting mathematical observation; it has deep implications for how circles scale. For instance, if you were to take any circle and double its diameter, the circumference would also double. Conversely, halving the diameter would result in halving the circumference. This direct proportionality means that once the diameter of a circle is known, its circumference can be predicted with complete accuracy. In practical terms, this scaling relationship has countless real-world applications. In fields like engineering, manufacturing, and architecture, where precise circular measurements are often required, the predictable scaling of the diameter and circumference simplifies the design process. It also illustrates a key property of Euclidean geometry, where relationships between geometric entities remain consistent and scale proportionally across all sizes. This concept of proportional scaling can be applied beyond simple geometric calculations to more complex systems, such as understanding how the rotational speed of an object is related to its size or how the surface area of spherical objects changes in relation to their diameter. This consistent, scalable relationship between diameter and circumference also leads to the development of other geometric formulas, such as those for the area of a circle (A=πr2A=πr2) or the volume of a sphere, and contributes to the unity of mathematical principles that govern circular shapes.

3. Historical Context and Practical Applications of the Diameter-Circumference Relationship in Various Fields:The relationship between the diameter and circumference has been fundamental not only in theoretical mathematics but also in its practical applications throughout history. Ancient civilizations, such as the Egyptians, Babylonians, and Greeks, sought to understand the nature of circles long before the advent of modern mathematics. The Egyptians used approximations of π to calculate the circumference of circular structures such as wells, while the Babylonians had their own early estimates of the constant. However, it was the Greeks, particularly the mathematician Archimedes, who made significant strides in understanding and approximating the value of pi. Archimedes famously used inscribed and circumscribed polygons to estimate the value of π with remarkable precision for his time, showing that the ratio between the diameter and circumference was a constant that could be approximated through increasingly accurate geometric methods. The idea that the circumference of a circle was directly tied to its diameter through this constant ratio laid the groundwork for future mathematical discoveries.

Fast-forwarding to the modern era, this relationship has profound implications in fields such as physics, engineering, astronomy, and even computer science. In rotational dynamics, for example, the diameter of a rotating object determines the distance traveled by a point on its circumference as it rotates, which is essential in understanding concepts like angular velocity and linear velocity. Similarly, in the construction of gears and wheels, the relationship between diameter and circumference is crucial in determining gear ratios, which affect the mechanical advantage in machines. In fluid mechanics, the formula for the circumference helps engineers design pipes, tanks, and other circular structures to ensure proper flow dynamics. In more advanced areas of physics, this relationship extends to phenomena like wave motion, where the circumference of circular paths (such as in a round wave or orbit) is tied to the fundamental periodic properties of the system.

Additionally, modern computing technologies, such as the design of circular or spherical components in devices, often rely on this simple but powerful relationship. This highlights how fundamental principles of geometry continue to have applications far beyond basic geometric problems, influencing everything from satellite orbit calculations to the construction of circular pathways in transportation networks. In essence, the simple relationship between diameter and circumference is not just a mathematical curiosity; it is the foundation upon which much of modern science, engineering, and technology has been built, demonstrating the timelessness and utility of this geometric principle.

This expanded explanation provides a deeper understanding of how the diameter and circumference of a circle are fundamentally related and the far-reaching implications of that relationship throughout history and in modern applications.

1. Proportionality Between Diameter and Circumference and Its Implications for Scaling Circles: A key aspect of the relationship is that the diameter and the circumference are always in a fixed ratio, no matter how large or small the circle is. This means that if you were to scale a circle by increasing or decreasing its diameter, its circumference would change proportionally. For instance, if the diameter of a circle is doubled, the circumference will also double. This proportional relationship underscores how the diameter determines the circle's overall size: a circle with a larger diameter will have a circumference that is predictably larger. This predictable scaling is one of the reasons why the relationship between diameter and circumference is so useful in geometry and various fields, such as engineering, physics, and even art, where precise circular measurements are needed. The concept of scaling circles by their diameters also relates directly to the formula for the area of a circle, A=πr2A=πr2, where the radius (half of the diameter) plays a similar role in determining the spatial area of the circle.

2. Historical and Practical Applications of the Diameter-Circumference Relationship: The concept of the diameter and its relationship to the circumference has had profound implications throughout history, especially in the development of mathematical and physical theories. The relationship was studied by ancient civilizations, including the Babylonians, Egyptians, and Greeks, each of whom approximated the value of π for practical use in construction, astronomy, and navigation. The formula C=π×dC=π×d was pivotal in shaping our understanding of circular motion, rotational dynamics, and other circular phenomena in physics, where the circumference of a circle plays a crucial role in determining things like angular velocity, rotational speed, and the distribution of forces. Today, this formula is foundational in fields ranging from satellite trajectory calculations to the design of round objects like wheels, gears, and pipes. The ability to calculate and predict the circumference based on the diameter has allowed for advances in engineering and technology, ensuring that circular objects meet precise specifications, particularly in fields such as manufacturing, where tolerances and measurements must be extremely accurate.

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Diameter is twice the radius:
The diameter (dd) of a circle is always twice the length of the radius (rr).
Formula:

d=2rd=2r

Radius is half the diameter:
The radius (rr) is half the length of the diameter (dd).
Formula:

r=d2r=2d​

Diameter is the longest chord:
The diameter is the longest possible straight line that can be drawn within the circle, passing through its center. The radius is the distance from the center to any point on the circle's edge, and the diameter is twice the radius.

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The diameter is twice the radius: The diameter (denoted as DD) of a circle is the longest straight line that can be drawn through the circle, passing through its center. The radius (denoted as rr) is the distance from the center of the circle to any point on its edge. The diameter is always twice the length of the radius, which means the formula for the diameter is D=2rD=2r.

The radius is half the diameter: Conversely, if you know the diameter of a circle, you can find the radius by dividing the diameter by 2. So, the formula for the radius is r=D2r=2D​. This shows that the radius is essentially half of the diameter.

Both measure the size of the circle but in different ways: The diameter measures the full width of the circle from one side to the other, passing through the center, while the radius measures the distance from the center to the edge. These two measurements are directly related, with the radius being the smaller, foundational measurement that determines the overall size of the circle. Understanding the relationship between the diameter and radius is key to calculating other properties of the circle, such as its circumference or area.

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Sure! Here are three key points about the relationship between the diameter and circumference of a circle:

1. Circumference is bigger than the diameter: The circumference is the distance all the way around the circle, and it's always longer than the diameter, which is just the straight line across the center.

2. Pi connects them: The circumference is about 3.14 times bigger than the diameter. This number is called pi (π).

3. Formula: You can find the circumference using the formula:
Circumference = π × Diameter.
So, if you know the diameter, just multiply it by 3.14 (or π) to get the circumference!

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The circumference is about 3.14 times the diameter.

If the diameter gets bigger, the circumference gets bigger in the same way.

The number 3.14 (π) is always the ratio between the circumference and diameter.

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First observation: circumference divided by diameter = 3.14159...

Second obeservation: Although the two numbers that are divided at the side are all different combanations when the points A and B are changed, when the fraction is simplified, it always equals to Pi.

Relationship: circumference divided by diameter always equal to Pi.

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Formula: The circumference of a circle is the distance all around the edge of the circle. To find it, you multiply the diameter (the distance across the circle through the center) by the special number ππ (which is about 3.14). So, the formula is:

Circumference=π×DiameterCircumference=π×Diameter

Fixed Relationship: The ratio (or relationship) between the circumference and the diameter is always the same, no matter how big or small the circle is. This ratio is always about 3.14, which is ππ.

Example: If you have a circle with a diameter of 10 cm, you can find the circumference by multiplying 10 cm by ππ(3.14). So, the circumference would be about 31.4 cm.

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The circumference of a circle is how far around it goes. It’s related to the diameter by the formula: Circumference = π × Diameter. This means if you multiply the diameter by about 3.14, you get the circumference.

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String roll->23cm

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1. Circumfrence divided by diameter is pi (3.14).

2. Whatever the size of the circle, the formula remains the same.

3. Diameter x Circumfrence =Pi

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Diameter equals to 9cm

Circumference=30cm

Length + diameter =39cm

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2 observations: we can change the radius and the diameter of the circle. We can see the angles degree.

The realtionships between the diameter and the circumference: The circumference, no matter what, all requires the diameter to be found out. Circumference= pi times diameter

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Firstly, I realise that no matter how long it is it when you hold the circle, it will always cover 3.14 of the diameter which is also pi. I also realised that the longer the diameter, the longer the conference of the circle. Thirdly, I also realised that the radius will also increase as diameter increases.

Relationship is stated inside my observations which is longer the diameter longer the radius it doesn't matter because the circumference divided by the diameter which is also the circumference divided by radius times two will always be 3. 14.

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Observations made:

1. As the diameter of a circle increases, the number of colours being unrolled stays the same.

2. As the diameter of a circle increases, the length of the unrolled circumference also increases.

3. The distance rolled by the circle after it completes one full rotation is equal to the length of its circumference.

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I observe that the circumference can fit three diameters,circumference divided by diameter is equal to pi, and there is no circumference that can be divide into a whole number by the diameter.

The relationship between the circumference and the diameter is that the circumference can always fit three diameters.

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Group 7 members: alexis lauren rebecca and amelia.

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Group 7 members: alexis lauren rebecca and amelia.

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Geogebra ( l )

As the diameter increases, the circumfrence of the circle increases. The distance rolled by the circle increases. ( idk how to explain )

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Photo 1

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Photo 2

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Photo 3

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Geogebra ( ll )

Circumference/Diameter=3.14259259259

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Diameter 2.4

Circumference 7.5

7.5/2.4=3.125

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Finding area of circle without using formula

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Finding area of circle without using formula

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List 2 ways how you can find the area of this circle without using the area of circle formula

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Show how you prove area of circle using area of rectangle formula.

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Show how you prove area of circle using area of rectangle formula.

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Download, type in and upload with your group's responses

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Colorful Box Digital Marketing Infographic Instagram Post (1)

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Is it true that the area of a circle is estimated to be 80% of area of square, when the diameter of the circle and length of side of the square is the same?

Upload your proof onto padlet.

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Math Around Us! Using knuckles to estimate area of circle

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Math Around Us! Using knuckles to estimate area of circle

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https://www.geogebra.org/m/AvyFBUh4

https://www.geogebra.org/m/fdp2YH6K

**Must play!

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https://www.geogebra.org/m/HRupB9hN

https://www.geogebra.org/m/WFbyhq9d

https://www.geogebra.org/m/TS4s4Aa5

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This is just an example. :)

So, my answer is: True, just not exactly (It's 78.53981634......).

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Is it true that the area of a circle is estimated to be 80% of area of square, when the diameter of the circle and length of side of the square is the same?

I agree with this statement. Explanation above ↑

Reminder: I think it is important to point about what estimated means. It does not mean that the area of the circle will be exactly 80%. However, when rounded off it will be estimated to be 80%!

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*I tried my best

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Area of Circle Reflection Sheet-Samantha Lee (25) 6R3

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I agree with this statement, since it is about

78.5398163397448...%, but it is estimated to be 80%

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Is it true that the area of a circle is estimated to be 80% of area of square, when the diameter of the circle and length of side of the square is the same?