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Chapter 29: Problem 787

Find the total length of the circumference of a circle of radius \(\mathrm{r}\).

Short Answer

Expert verified
The total length of the circumference of a circle with radius r is given by the formula \(C = 2\pi r\).

Step by step solution

01

Write down the formula for the circumference of a circle

The formula for the circumference of a circle is given by: C = 2蟺r, where C is the circumference and r is the radius.
02

Plug in the given radius

We're given a circle with radius r. We need to plug this value into the circumference formula: C = 2蟺(r).
03

Calculate the total length of the circumference

To find the total length of the circumference, we just need to perform the multiplication: C = 2蟺r. So, the total length of the circumference of the circle with radius r is 2蟺r.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometry
Geometry is a fascinating area of mathematics that deals with the study of shapes, sizes, and the properties of space.
By understanding geometry, we can solve problems related to the physical world around us. When it comes to circles, geometry helps us analyze and calculate various characteristics such as the circumference, area, or the position of points in circles.
  • Geometry involves the theoretical understanding of figures, which are the forms and shapes we see.
  • It also encompasses principles, helping us measure and calculate the physical dimensions of objects.
In the context of our problem, the primary focus is understanding how a key geometric concept鈥攖he circumference鈥攔elates to a basic shape, the circle.
Circle
A circle is one of the simplest and most familiar shapes in geometry. It is defined as the set of all points in a plane that are equidistant from a given point called the center.
This distance is known as the radius, and it's crucial in understanding the relationship between various aspects of a circle, like its circumference and area.
  • The circumference of a circle is the total distance around the circle.
  • The concept of circles is ubiquitous; circles are found in everything from wheels to clocks and even planets.
Understanding circles is essential for tackling a wide range of real-world problems, and they often appear in mathematical and scientific contexts.
Radius
The radius of a circle is a key component when exploring its properties. It is the distance from the center to any point on the circle itself.
The length of the radius determines the size of the circle. A larger radius means a bigger circle, while a smaller radius equates to a smaller circle.
  • The radius is used in formulas to calculate important circle characteristics like the circumference and area.
  • For any circle, the radius is constant, meaning it doesn't change regardless of which point on the circle you measure to from the center.
In calculations, the radius is often symbolized as \( r \), and knowing its length is key to finding the circumference, which is calculated as \( 2\pi r \).
Pi (蟺)
Pi (蟺) is a special mathematical constant used across various fields of mathematics and science. It represents the ratio of a circle's circumference to its diameter.
Pi is an irrational number, which means it cannot be expressed as a simple fraction and has a non-repeating decimal form, often approximated as 3.14159.
  • Pi is critical in calculations involving circles, appearing in formulas for circumference and area.
  • Despite its complexity, Pi helps simplify the process of finding the circumference through the formula \( C = 2\pi r \).
The symbol \( \pi \) is widely recognized in mathematics and is considered one of the most important constants, connecting geometry with algebra and several other areas of math.

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