A circle can be divided into four congruent arcs that measure

$\displaystyle \frac{1}{4} \cdot 360 ^{\circ } = 90 ^{\circ }$.

If the four (congruent) chords of the arcs are constructed, they will form a square with sides of length $\displaystyle x + 2$. The diagonal of a square has length $\displaystyle \sqrt{2}$ times that of a side, which will be

$\displaystyle (x + 2) \sqrt{2} = x \sqrt{2} + 2 \sqrt{2}$

A diagonal of the square is also a diameter of the circle; the circle will have radius half this length, or

$\displaystyle \frac{1}{2} \left ( x \sqrt{2} + 2 \sqrt{2} \right ) =\frac{1}{2} x \sqrt{2} + \sqrt{2}$

The area of a circle with radius $\displaystyle r$ is $\displaystyle A = \pi r^{2}$.

Let $\displaystyle R$ be the radius of the smaller circle. Its area is $\displaystyle A_{1} = \pi R^{2}$. The area of the larger circle is $\displaystyle A_{2} = \pi \cdot 10^{2} = 100 \pi$. Since the area of the region between the circles is $\displaystyle 50 \pi$, and is the difference of these areas, we have

$\displaystyle 100 \pi - \pi R^{2} = 50 \pi$

$\displaystyle 100 \pi - \pi R^{2} - 50 \pi + \pi R^{2} = 50 \pi - 50 \pi + \pi R^{2}$

$\displaystyle 50 \pi = \pi R^{2}$

$\displaystyle R^{2} = 50$

$\displaystyle R = \sqrt{50} = \sqrt {25 } \cdot \sqrt{2} = 5 \sqrt{2}$

The smaller circle has radius $\displaystyle 5 \sqrt{2}$.

A $\displaystyle 60^{\circ }$ arc of a circle is $\displaystyle \frac{60}{360 } = \frac{1}{6}$ of the circle. Since the length of this arc is $\displaystyle x+ 4$, the circumference is $\displaystyle \textup{six times }$ this, or

$\displaystyle C = 6( x+ 4) = 6x+ 24$

The radius of a circle is its circumference divided by $\displaystyle 2 \pi$; therefore, the radius is

$\displaystyle r = \frac{C}{2 \pi} = \frac{6x+24}{2\pi} = \frac{3}{\pi}x+\frac{12}{\pi}$

A circle can be divided into $\displaystyle 6$ congruent arcs that measure

$\displaystyle \frac{1}{6} \cdot 360 ^{\circ } = 60 ^{\circ }$.

If the $\displaystyle 6$ (congruent) chords are constructed, the figure will be a regular hexagon. The radius of this hexagon will be equal to the length of one side - one $\displaystyle 60 ^{\circ }$ chord of the circle; this radius will coincide with the radius of the circle. Therefore, the radius of the circle is the length of chord $\displaystyle \overline{AB}$, or $\displaystyle x+2$.

If a monster truck's wheels have circumference of $\displaystyle 31.4 ft$,  what is the distance from the ground to the center of the wheel?

This question is asking us to find the radius of a circle. the distance from the outside of the circle to the center is the radius. We are given the circumference, so use the following formula:

$\displaystyle Circ=2 \pi r$

Then, plug in what we know and solve for r

$\displaystyle 31.4=2 \pi r$

$\displaystyle r=\frac{31.4}{2 \pi}=4.997\approx5ft$

The area of a circle with radius $\displaystyle r$ is $\displaystyle A = \pi r^{2}$.

Let $\displaystyle R$ be the radius of the larger circle. Its area is $\displaystyle A_{1} = \pi R^{2}$. The area of the smaller  circle is $\displaystyle A_{2} = \pi \cdot 10^{2} = 100 \pi$. Since the area of the region between the circles is $\displaystyle 50 \pi$, and is the difference of these areas, we have

$\displaystyle \pi R^{2} - 100 \pi = 50 \pi$

$\displaystyle \pi R^{2} - 100 \pi+ 100 \pi = 50 \pi + 100 \pi$

$\displaystyle \pi R^{2} = 150 \pi$

$\displaystyle R^{2} = 150$

$\displaystyle R = \sqrt{150} = \sqrt {25 } \cdot \sqrt{6} = 5 \sqrt{6}$

The smaller circle has radius $\displaystyle 5 \sqrt{6}$.