Find the area of a circle with a circumference of 31.4 units. units \( ^{2} \)

To find the area of the circle, we first need to determine its radius. The circumference \( C \) of a circle is given by the formula \( C = 2\pi r \). Given \( C = 31.4 \) units, we can rearrange the formula to solve for \( r \): \[ r = \frac{C}{2\pi} = \frac{31.4}{2\pi} \approx 5 \text{ units} \] Now that we have the radius, we can find the area \( A \) using the formula \( A = \pi r^2 \): \[ A = \pi (5)^2 = 25\pi \approx 78.54 \text{ units}^2 \] So, the area of the circle is approximately \( 78.54 \) square units. Did you know that circles have fascinated humans since ancient times? The ancient Greeks studied their properties extensively, with mathematicians like Euclid and Archimedes making significant contributions to our understanding of geometry. Their work laid the groundwork for many mathematical principles we still use today! Circles aren't just confined to math; they're all around us in the real world! For instance, engineers utilize them in the design of wheels and gears, while architects often incorporate circular arches in buildings for both aesthetic appeal and structural integrity. Not to mention, the simple joy of a pizza slice, starting from a perfect circle!