Understanding the Context

The formula for the circumference ( C ) of a circle is given by:
[ C = 2\pi r ]
where:
- ( C ) is the circumference,
- ( \pi ) (pi) is a mathematical constant approximately equal to 3.14,
- ( r ) is the radius, the distance from the center of the circle to its outer edge.

Rearranging this equation to solve for ( r ), we get:
[ r = rac{C}{2\pi} ]

This rearrangement allows us to calculate the radius when the circumference is known—partically useful when working with circular objects or structures.

Plugging in the Numbers

Key Insights

Given ( C = 31.4 , \ ext{meters} ), substitute into the rearranged formula:
[ r = rac{31.4}{2\pi} ]

Using ( \pi pprox 3.14 ):
[ r = rac{31.4}{2 \ imes 3.14} = rac{31.4}{6.28} = 5 , \ ext{meters} ]

This confirms that ( r = 5 , \ ext{meters} ), offering a precise and simple way to determine the radius from the circumference.

Why This Calculation Matters

This formula is widely applied across disciplines:

Final Thoughts

• Construction and architecture: Determining circular foundations, columns, or domes.
  - Manufacturing: Designing round gears, pipes, and mechanical components.
  - Earth sciences: Estimating the radius of circular craters, ponds, or storm systems.
  - Education: Teaching fundamental principles of geometry and trigonometry.

Understanding how to solve for ( r ) using the circumference formula empowers accurate measurement and problem solving in practical situations.

Step-by-Step Breakdown

To simplify the calculation:
1. Start with ( r = rac{C}{2\pi} ).
2. Substitute ( C = 31.4 ) and ( \pi pprox 3.14 ):
[ r = rac{31.4}{2 \ imes 3.14} ]
3. Multiply the denominator: ( 2 \ imes 3.14 = 6.28 ).
4. Divide:
[ r = rac{31.4}{6.28} = 5 ]
Thus, ( r = 5 , \ ext{meters} ).

This straightforward calculation demonstrates how mathematical constants combine to decode real-world measurements efficiently.

Final Thoughts