The radius of a circle with diameter 18 centimeters is half that, or 9 centimeters. The area of a $\displaystyle 60^{\circ }$ sector of the circle is 

$\displaystyle A = \frac{60}{360 }\cdot \pi \cdot 9^{2} =\frac{1}{6} \cdot \pi \cdot 81 \approx 42.4 \textrm{ cm}^{2}$

The equation to find the area of a sector is $\displaystyle (\frac{x}{360})\Pi r^2$.

Substitute the given radius in for $\displaystyle r$ and the given angle in for $\displaystyle x$ to get:

$\displaystyle (\frac{120}{360})\Pi (3^2)$

Simplify the equation to get the area:

$\displaystyle (\frac{1}{3})\Pi9=3\Pi$ 

To find the area of the shaded region, you must subtract the area of the triangle from the area of the sector.

The formula for the shaded area is:

$\displaystyle A=A_{sector}-A_{triangle}$

$\displaystyle A = \pi(r^2)(part)-\frac{1}{2}(b)(h)$,

where $\displaystyle r$ is the radius of the circle, $\displaystyle part$ is the fraction of the sector, $\displaystyle b$ is the base of the triangle, and $\displaystyle h$ is the height of the triangle.

In order to the find the base and height of the triangle, use the formula for a $\displaystyle 30-60-90$ triangle:

$\displaystyle a-a\sqrt{3}-2a$, where $\displaystyle a$ is the side opposite the $\displaystyle \measuredangle30$.

Plugging in our final values, we get:

$\displaystyle A = \pi(18m^2)(\frac{120}{360})-\frac{1}{2}(18\sqrt{3}m)(9m)$

$\displaystyle A = 108\pi-81\sqrt{3}m^2$

The formula for the area of a sector is

$\displaystyle A = \pi r^2 (part)$,

where $\displaystyle r$ is the radius of the circle and $\displaystyle part$ is the fraction of the sector.

Plugging in our values, we get:

$\displaystyle A = \pi (6\ m)^2 (\frac{90}{360})$

$\displaystyle A=9 \pi\ m^2$

Area of Circle = πr² = π4² = 16π

Total degrees in a circle = 360

Therefore 45 degree slice = 45/360 fraction of circle = 1/8

Shaded Area = 1/8 * Total Area = 1/8 * 16π = 2π

We know that the right angle rests at the center of the circle; thus, the sides of the triangle represent the radius of the circle.

$\displaystyle r=6$

Because the sector of the circle is defined by a right triangle, the region corresponds to one-fourth of the circle.

$\displaystyle \frac{degrees\ in\ angle}{degrees\ in\ circle}=\frac{90^o}{360^o}=\frac{1}{4}$

First, find the total area of the circle and divide it by four to find the area of the depicted sector.

$\displaystyle \small A(circle)=\pi(r)^2$

$\displaystyle \small A(sector)=\frac{\pi(r)^2}{4}$

$\displaystyle \small A(sector)=\frac{\pi(6)^2}{4}=\frac{\pi(36)}{4}=9\pi$

Next, calculate the area of the triangle.

$\displaystyle \small A(tri)=\frac{1}{2}bh=\frac{1}{2}(6)(6)=18$

Finally, subtract the area of the triangle from the area of the sector.

$\displaystyle \small A(shaded)=9\pi-18$

To solve this, we must begin by finding the area of the diagram, which is the area of the square less the area of the semicircle. 

The area of the square is straightforward:

30 * 30 = 900 square feet

Because each side is 30 feet long, AB + BC + CD = 30.  

We can substitute BC for AB and CD since all three lengths are the same:

BC + BC + BC = 30 

3BC = 30

BC = 10

Therefore the diameter of the semicircle is 10 feet, so the radius is 5 feet.

The area of the semi-circle is half the area of a circle with radius 5.  The area of the full circle is 5²π = 25π, so the area of the semi-circle is half of that, or 12.5π.

The total area of the plot is the square less the semicircle: 900 - 12.5π square feet

The cost of upkeep is therefore 2.5 * (900 – 12.5π) = $(2250 – 31.25π).

First, we figure out what fraction of the circle is contained in sector OPQ: $\displaystyle \frac{30^{\circ}}{360^{\circ}}= \frac{1}{12}$, so the total area of the circle is $\displaystyle \dpi{100} \small 12\times 3\pi=36$ .

Using the formula for the area of a circle, $\displaystyle {\pi}r^{2}$, we can see that $\displaystyle \dpi{100} \small r=6$.

We can use this to solve for the circumference of the circle, $\displaystyle 2{\pi}r$, or $\displaystyle 12{\pi}$.

Now, OP and OQ are both equal to r, and PQ is equal to $\displaystyle \dpi{100} \small \frac{1}{12}$ of the circumference of the circle, or $\displaystyle {\pi}$.

To get the perimeter, we add OP + OQ + PQ, which give us $\displaystyle 12+{\pi}$.

If a quarter of the area of a circle is $\displaystyle 9\pi$, then the area of the whole circle is $\displaystyle 36\pi$.  This means that the radius of the circle is 6.  The diameter is 12.  Thus, the circumference of the circle is $\displaystyle 12\pi$.  One fourth of the circumference is $\displaystyle 3\pi$.