## Task 3

On a school playground, a large circle is drawn. The center of the circle is point $O$, and points $A, B$, and $C$ are marked on the circumference. Angles $\measuredangle A O B$ and $\measuredangle B O C$ are acute, and the lengths $\overline{O A}$ and $\overline{O C}$ are perpendicular to each other. The part of the circle bounded by segments $\overline{O A}, \overline{O B}$, and the arc $\widehat{A B}$ is painted green, and the part of the circle bounded by segments $\overline{O B}, \overline{O C}$, and the arc $\widehat{B C}$ is painted orange. Ivica measured with his feet that the circumference of the circle is 240 feet and that the perimeter of the green shape is 18 feet larger than the perimeter of the orange shape. If the area of the entire circle is $72 \mathrm{~m}^{2}$, what is the area of the orange shape?

![](https://cdn.mathpix.com/cropped/2024_05_30_22f201ffe2378dbba52dg-3.jpg?height=246&width=282&top_left_y=1339&top_left_x=1635)

Note: The perimeter of a geometric shape is the length of its boundary.
## Solution:

The orange and green circular segments together form a quarter of the circle bounded by an arc (or circular arc) of length $240: 4=60$ feet.

The green segment is bounded by two radii and the circular arc $\widehat{A B}$, and the orange segment by two radii and the circular arc $\widehat{B C}$.

The perimeter of the green segment is $2 \mathrm{r}+|\widehat{A B}|$, and the perimeter of the orange segment is $2 \mathrm{r}+|\widehat{B C}|$. The perimeter of the green segment is 18 feet greater, so we have

$2 \mathrm{r}+|\widehat{B C}|+18=2 \mathrm{r}+|\widehat{A B}|$. From this, it follows that $|\widehat{B C}|+18$ feet $=|\widehat{A B}|$.

![](https://cdn.mathpix.com/cropped/2024_05_30_22f201ffe2378dbba52dg-3.jpg?height=574&width=560&top_left_y=1775&top_left_x=1476)

Since both circular arcs together form a quarter of the circle,

we have $2|\widehat{B C}|+18$ feet $=60$ feet, or $|\widehat{B C}|=21$ feet, $|\widehat{A B}|=39$ feet.

The area of the circle is $72 \mathrm{~m}^{2}$, so the area of the circular segment corresponding to an arc of 1 foot is $72: 240=0.3 \mathrm{~m}^{2}$.

The area of the orange circular segment is $21 \cdot 0.3=6.3 \mathrm{~m}^{2}$.