Problem 3 - Precalculus : Building Concepts and Connections

Question

Find the circumference of each circle given its radius or diameter. Leave your answer in terms of \(\pi .\) diameter 12 inches

Step-by-Step Solution

Answer

The circumference of the circle is \(12\pi\) inches.

Step 1: Understand the problem and the formula for circumference

We are given the diameter of the circle, and are asked to find the circumference of the circle in terms of \(\pi\). We will use the formula for circumference, which is \(C = \pi d\), where \(C\) represents the circumference of the circle and \(d\) represents the diameter of the circle.

Step 2: Substitute the given diameter into the formula

Substitute \(d = 12\) inches into the circumference formula. That will give us \(C = \pi \times 12\).

Step 3: Calculate the circumference

Multiplying \(\pi\) by 12, we get \(C = 12\pi\) inches.

Key Concepts

Circumference

The circumference of a circle is a fundamental concept to grasp when studying circles. It is the total distance around the circle, similar to the perimeter of a polygon. Understanding circumference helps in numerous real-world applications like designing wheels, tracks, and even in geographical mapping. The idea of circumference provides a way to measure how "big" a circle is in terms of its round boundary, making it an essential tool for various mathematical and engineering calculations.

Diameter

Diameter is a crucial element in understanding circles, as it is directly related to the circumference. The diameter of a circle is the straight line passing through the center, connecting two points on the circle's boundary. It represents the longest distance across the circle. Knowing the diameter is essential, as it allows one to calculate the circumference easily. In simple terms, if you have a circle, and you know the diameter, it essentially acts as a key to unlock information about the circle's extent and size.

Circumference Formula

The circumference formula is a simple yet powerful mathematical expression used to find the boundary line of a circle. There are two primary formulas for the circumference depending on whether you have the diameter or the radius:

• Using the diameter: \( C = \pi d \)
• Using the radius: \( C = 2\pi r \)

Both formulas are designed to work with the value of \( \pi \), an irrational number approximately equal to 3.14159. This mathematical constant provides the ratio of any circle's circumference to its diameter, making these formulas universally applicable in geometry and beyond. By using these formulas, one can efficiently find the circumference as shown in the original exercise where the diameter was 12 inches, leading to the circumference \( 12\pi \) inches.