Knowing the perimeter (or circumference) of a circle allows you to easily calculate its radius using algebra. This article will guide you through the process using the formula \( P = 2 \cdot \pi \cdot r \). We will explain the formula and provide a step-by-step example to illustrate the calculations.
The Formula for the Perimeter of a Circle
The perimeter \( P \) of a circle is given by:
\[ P = 2 \cdot \pi \cdot r \]
Where:
- \( P \) is the perimeter (circumference) of the circle.
- \( \pi \) (Pi) is a constant approximately equal to 3.14159.
- \( r \) is the radius of the circle.
Explanation of the Formula
1. **\( 2 \cdot \pi \cdot r \)**: This formula represents the relationship between the radius of the circle and its circumference. Multiplying the radius by \( 2 \cdot \pi \) gives the total length around the circle.
Isolating the Radius \( r \)
To determine the radius \( r \), we need to isolate \( r \) in the formula.
\[ P = 2 \cdot \pi \cdot r \]
Divide both sides of the equation by \( 2 \cdot \pi \):
\[ r = \frac{P}{2 \cdot \pi} \]
Step-by-Step Calculation
Let's work through an example to illustrate the process.
Example:
Suppose we have a circle with a perimeter \( P = 31.4159 \) units. We want to find the radius of the circle.
Step 1: Identify the Given Value
Given:
- Perimeter \( P = 31.4159 \) units
Step 2: Substitute the Given Value into the Formula
\[ r = \frac{31.4159}{2 \cdot \pi} \]
Step 3: Calculate the Radius
First, calculate \( 2 \cdot \pi \) (approximated as 3.14159):
\[ 2 \cdot \pi = 2 \cdot 3.14159 \approx 6.28318 \]
Next, divide the perimeter by this value:
\[ r = \frac{31.4159}{6.28318} \]
\[ r \approx 5 \]
Final Value
For a circle with a perimeter of 31.4159 units, the radius is approximately 5 units.
