Problem 11

Question

Find the circumference of the circle with the given radius or diameter. $$d=23.1 \mathrm{mm}$$

Step-by-Step Solution

Verified

Answer

The circumference is approximately 72.57 mm.

1Step 1: Understanding the Problem

We need to find the circumference of a circle, which is the distance around it. The problem gives us the diameter of the circle as 23.1 mm.

2Step 2: Using the Circumference Formula

The formula to find the circumference of a circle when the diameter is given is: \[ C = \pi \times d \]where $C$ is the circumference and $d$ is the diameter.

3Step 3: Substituting the Diameter Value

Plug the given diameter into the formula: \[ C = \pi \times 23.1 \text{ mm} \]

4Step 4: Calculating with Approximation of Pi

Using the approximation $ \pi \approx 3.14159 $, calculate \[ C = 3.14159 \times 23.1 \text{ mm} \approx 72.566529 \text{ mm} \]

5Step 5: Rounding to an Appropriate Precision

Round the result to two decimal places for clarity. Hence, the circumference is \[ C \approx 72.57 \text{ mm} \].

Key Concepts

Understanding DiameterConcept of a CircleUnderstanding Pi ($\pi$)

Understanding Diameter

The diameter of a circle is a critical component when dealing with circular measurements. It is defined as the straight line that passes from one side of the circle to the other, passing through the center. This line essentially divides the circle into two equal halves.

The diameter is always twice the length of the radius. The formula that links the diameter ($d$) and radius ($r$) is: $d = 2r$.
When you know the diameter of a circle, you can easily find the circumference by using the circumference formula.

Understanding the diameter helps us see how it sets the boundary for the circle's size. It's the longest straight distance you can measure in a circle, making it essential for calculations involving circles.By knowing the diameter, we can proceed to apply it in understanding other properties, like the circumference.

Concept of a Circle

A circle is a simple, yet fascinating geometric shape defined by its round form. All points on the boundary of a circle are equidistant from a fixed point called the center.

The distance from this center to any point on the circle is the radius.
Each circle has a unique property of being perfectly symmetrical around its center.

Circles are the foundation of understanding rotational symmetry and appear in various real-world applications, from the wheels of a car to the face of a clock. The properties of a circle, like the radius, diameter, and circumference, are inherently related, which simplifies calculations and understanding even for complex geometric problems.
Circles are everywhere, and recognizing their properties enables us to navigate through geometry with ease and confidence.

Understanding Pi ($\pi$)

Pi ($\pi$) is one of the most renowned constants in mathematics, essential for calculations involving circles. Pi represents the ratio of the circumference of any circle to its diameter.

Pi is an irrational number, which means that its digits go on forever without repeating.
The commonly used approximation of $\pi$ is 3.14159, though it extends infinitely without a predictable pattern.

Pi is foundational in equations dealing with the circle, making it indispensable in both practical and theoretical geometry. In formulae, when the diameter ($d$) is known, finding the circumference ($C$) becomes straightforward with $C = \pi \times d$.
Understanding Pi helps us appreciate the precision needed in circular calculations, allowing us to handle them with accuracy, whether in academic problems or real-life scenarios like engineering and architecture.

Problem 10

Problem 12