Sector TYP occupies 43% of a circle. Find the degree measure of angle TYP.

Use the following formula and solve for x:

$\displaystyle \frac{x}{360^{\circ}}100=43\%$

Begin by dividing over the 100

$\displaystyle \frac{x}{360^{\circ}}=.43$

Then multiply by 360

$\displaystyle x=.43*360^{\circ}=154.8^{\circ}$

If sector AJL covers 45% of circle J, what is the measure of sector AJL's central angle?

To find an angle measure from a percentage, simply convert the percentage to a decimal and then multiply it by 360 degrees.

$\displaystyle 45\%\rightarrow 0.45$

$\displaystyle 0.45 * 360^{\circ}=162^{\circ}$

So, our answer is 162 degrees.

If we let $\displaystyle C$ be the circumference of the circle, then the length of $\displaystyle \overarc{AB}$ is $\displaystyle \frac{80}{360} = \frac{2}{9 }$ of the circumference, so

$\displaystyle \frac{2}{9 } C = 12 \pi$

$\displaystyle \frac{9 }{2}\cdot \frac{2}{9 } C =\frac{9 }{2}\cdot 12 \pi$

$\displaystyle C =54\pi$

The radius is the circumference divided by $\displaystyle 2 \pi$:

$\displaystyle r= 54 \pi \div 2 \pi = 27$

Use the formula to find the area of the entire circle:

$\displaystyle A = \pi r ^{2} = \pi \left (27 \right )^{2} = 729 \pi$

The area of the white region is $\displaystyle 1 - \frac{2}{9} = \frac{7}{9}$ of that of the circle, or 

$\displaystyle \frac{7}{9} \cdot 729 \pi = 567 \pi$

The radius of a circle is found by dividing the circumference $\displaystyle C = 30 \pi$ by $\displaystyle 2 \pi$:

$\displaystyle r= \frac{C}{2 \pi}= \frac{30 \pi}{2 \pi} = 15$

The area of the entire circle can be found by substituting for $\displaystyle r$ in the formula:

$\displaystyle A = \pi r ^{2} = \pi \cdot 15 ^{2} = 225 \pi$.

The area of the shaded $\displaystyle 80 ^{\circ }$ sector is $\displaystyle \frac{80 }{360 }$ of the total area:

$\displaystyle \frac{80 }{360 } \times 225 \pi = \frac{2}{9} \times 225 \pi = 50 \pi$

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of $\displaystyle 65^{\circ}$. If the screen has a radius of 4 inches, what is the area of the highlighted wedge?

To begin, let's recall our formula for area of a sector.

$\displaystyle A_{sector}=\frac{\theta}{360^{\circ}}\pi r^2$

Now, we have theta and r, so we just need to plug them in and simplify!

$\displaystyle A_{sector}=\frac{65^{\circ}}{360^{\circ}}\pi (4in)^2=2.89\pi in^2$

So our answer is $\displaystyle 2.89\pi in^2$

The distance that the tip of the minute hand moves during one hour is the circumference of a circle with radius 6 feet. This circumference is $\displaystyle 2\pi r = 2 \pi \times 6 = 12\pi$ feet. One hour and twenty minutes is $\displaystyle 1 \frac{1}{3}$ hours, so the tip of the hand moved $\displaystyle 1 \frac{1}{3} \times 12 \pi = 16 \pi$ feet, or $\displaystyle 16 \pi \times 12 = 192 \pi$ inches.

The distance that the tip of the minute hand moves during one hour is the circumference of a circle with radius $\displaystyle 3$ feet. This circumference is $\displaystyle 2\pi r = 2 \pi \times 3 = 6\pi$ feet. $\displaystyle 20$ minutes is one-third of an hour, so the tip of the minute hand moves $\displaystyle \frac{1}{3} \times 6\pi = 2\pi$ feet, or $\displaystyle 2\pi \times 12 = 24\pi$ inches.

The measure of an arc - $\displaystyle \overarc{AB}$ - intercepted by an inscribed angle - $\displaystyle \angle ACB$ - is twice the measure of that angle, so

$\displaystyle m \overarc{AB} = 2 \cdot m \angle ACB$

$\displaystyle 4y + 13 = 2 (2x- 7)$

$\displaystyle 4y + 13 = 4x- 14$

$\displaystyle 4y + 13 - 13 = 4x- 14 - 13$

$\displaystyle 4y = 4x- 27$

$\displaystyle \frac{4y}{4} = \frac{4x- 27}{4}$

$\displaystyle y = x- \frac{27}{4}$

The circumference of a circle is $\displaystyle 2 \pi$ multiplied by its radius , so

$\displaystyle C = 2 \pi \cdot OA = 2 \pi \cdot 12 = 24 \pi$.

$\displaystyle \angle ACB$, being an inscribed angle of the circle, intercepts an arc $\displaystyle \overarc {AB}$ with twice its measure:

$\displaystyle m \overarc {AB} = 2 \cdot m\angle ACB = 2 \cdot 72 ^{\circ } = 144 ^{\circ }$

The length of $\displaystyle \overarc {AB}$ is the circumference multiplied by $\displaystyle \frac{144}{360}$:

$\displaystyle \frac{144}{360} \times 24 \pi = \frac{48 \pi }{5}$.

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of $\displaystyle 65^{\circ}$. If the screen has a radius of 4 inches, what is the length of the arc of the highlighted wedge?

To begin, let's recall our formula for length of an arc.

$\displaystyle Arc=\frac{\theta}{360^{\circ}}2 \pi r$

Now, just plug in and simplify

$\displaystyle Arc=\frac{65^{\circ}}{360^{\circ}}2 \pi (4in)=\frac{13^{\circ}}{9^{\circ}}\pi in\approx 4.54in$

So, our answer is 4.54in