Sectors

Help Questions

Math › Sectors

Questions 1 - 10

Find the area of the shaded region:

Screen_shot_2014-02-27_at_6.35.30_pm

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

$108\pi-81\sqrt{2}m^2$

$81\pi-81\sqrt{3}m^2$

$108\pi-108\sqrt{3}m^2$

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

Explanation

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:

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

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

where is the radius of the circle, is the fraction of the sector, is the base of the triangle, and is the height of the triangle.

In order to the find the base and height of the triangle, use the formula for a triangle:

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

Plugging in our final values, we get:

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

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

Find the area of the following sector:

$9 \pi\ m^2$

$18 \pi\ m^2$

$6 \pi\ m^2$

$3 \pi\ m^2$

$12 \pi\ m^2$

Explanation

The formula for the area of a sector is

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

where is the radius of the circle and is the fraction of the sector.

Plugging in our values, we get:

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

$A=9 \pi\ m^2$

Find the length of the arc if the radius of a circle is and the measure of the central angle is degrees.

$\frac{11}{3}\pi$

$4\pi$

$\frac{13}{3}\pi$

$5\pi$

Explanation

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

$\text{Length of Arc}=\frac{\text{Measure of central angle}}{360}\times2\pi\times \text{radius}$

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{55}{360}\times 2\pi \times 12$

Solve.

$\text{Length of Arc}=\frac{11}{3}\pi$

In the figure below, $\text{chord AB}\parallel\text{chord CD}$ . If $\text{Arc AC}$ is degrees, in degrees, what is the measure of $\text{Arc BD}$ ?

The measurement of $\text{Arc BD}$ cannot be determined with the information given.

Explanation

Recall that when chords are parallel, the arcs that are intercepted are congruent. Thus, $\text{Arc AC}=\text{Arc BD}$ .

Then, $\text{Arc BD}$ must also be degrees.

Arcs

$m \angle AXC = 135^{\circ }$ ; $arc\ \widehat{AB} = 24^{\circ }$ ; $arc\ \widehat{AC} = 94^{\circ }$

Find the degree measure of $arc\ \widehat{CD}$ .

$66^{\circ }$

Not enough information is given to answer this question.

$164^{\circ}$

$82^{\circ}$

$132^{\circ }$

Explanation

When two chords of a circle intersect, the measure of the angle they form is half the sum of the measures of the arcs they intercept. Therefore,

$m \angle AXB = \frac{m; \widehat{AB} + m; \widehat{CD} }{2}$

Since $\angle AXB$ and $\angle AXC$ form a linear pair, $m \angle AXB + m \angle AXC = 180$ , and $m \angle AXB = 180 - m \angle AXC = 180 - 135 = 45^{\circ }$ .

Substitute $m; \widehat{AB} = 24$ and $m \angle AXB = 45$ into the first equation:

$45 = \frac{24 + m; \widehat{CD} }{2}$

$90 =24 + m; \widehat{CD}$

$66 = m; \widehat{CD}$

A sector in a circle with a radius of has an area of $\frac{1}{4}\pi$ . In degrees, what is the measurement of the central angle of the sector?

$5.625^\circ$

$2.775^\circ$

$6.225^\circ$

$8.555^\circ$

Explanation

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times \frac{1}{4}\pi}{\pi\times 4^2}=\frac{90\pi}{16\pi}=5.625$

The central angle is degrees.

Find the area of the sector that has a central angle of degrees and a radius of .

Explanation

The circle in question can be drawn as shown by the figure below:

Since the area of a sector is just a fractional part of the area of a circle, we can write the following equation to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$ , where is the radius of the circle.

Plug in the given central angle and radius to find the area of the sector.

$\text{Area of Sector}=\frac{160}{360}\pi(12)^2=201.06$

Make sure to round to two places after the decimal.

Find the area of a sector with a central angle of degrees and a radius of .

Explanation

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}=\frac{\text{Central Angle}}{360}\times\pi\times r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}=\frac{124}{360}\times\pi\times 12^2$

Solve and round to two decimal places.

$\text{Area of Sector}=155.82$

Find the area of a sector if it has an arc length of and a radius of .

$\frac{3}{2}$

The area of the sector cannot be determined.

$\frac{5}{2}$

$\frac{7}{2}$

Explanation

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$