Sectors - Math

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Question

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

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

Answer

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:

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

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

$66 = m; \widehat{CD}$

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Question

A sector comprises 20% of a circle. What is the central angle of the sector?

Answer

Proporations can be used to solve for the central angle. Let equal the angle of the sector.

$\frac{20}{100}=\frac{x}{360}$

Cross mulitply:

Solve for :

$\frac{7200}{100}=\frac{100x}{100}$

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Question

The radius of the circle above is and $\angle A=45^{\circ}$ . What is the area of the shaded section of the circle?

Answer

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

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Question

is a square.

$\overline{AB}=\overline{BC}=\overline{CD}$

The arc from to is a semicircle with a center at the midpoint of $\overline{BC}$ .

All units are in feet.

The diagram shows a plot of land.

The cost of summer upkeep is $2.50 per square foot.

In dollars, what is the total upkeep cost for the summer?

Answer

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).

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Question

A circular, 8-slice pizza is placed in a square box that has dimensions four inches larger than the diameter of the pizza. If the box covers a surface area of 256 in2, what is the surface area of one piece of pizza?

Answer

The first thing to do is calculate the dimensions of the pizza box. Based on our data, we know 256 = s2. Solving for s (by taking the square root of both sides), we get 16 = s (or s = 16).

Now, we know that the diameter of the pizza is four inches less than 16 inches. That is, it is 12 inches. Be careful! The area of the circle is given in terms of radius, which is half the diameter, or 6 inches. Therefore, the area of the pizza is π * 62 = 36π in2. If the pizza is 8-slices, one slice is equal to 1/8 of the total pizza or (36π)/8 = 4.5π in2.

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Question

In the figure, PQ is the arc of a circle with center O. If the area of the sector is $3\pi$ what is the perimeter of sector?

Answer

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

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

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

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

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

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Question

If B is a circle with line AC = 12 and line BC = 16, then what is the area formed by DBE?

Answer

Line AB is a radius of Circle B, which can be found using the Pythagorean Theorem:

$AB^2=AC^2+BC^2\rightarrow AB=\sqrt{AC^2+BC^2}=\sqrt{16^2+12^2}=\sqrt{400}=20$

Since AB is a radius of B, we can find the area of circle B via:

$Area=\pi R^2=\pi(20^2)=400\pi$

Angle DBE is a right angle, and therefore $\frac{1}{4}$ of the circle so it follows:

$Area(DBE)=\frac{400}{4}\pi=100\pi$

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Question

Find the area of the shaded segment of the circle. The right angle rests at the center of the circle.

Answer

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.

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

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

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

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

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

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

Next, calculate the area of the triangle.

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

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

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

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Question

To the nearest tenth, give the area of a $60^{\circ }$ sector of a circle with diameter 18 centimeters.

Answer

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

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

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Question

Find the area of a sector that has an angle of 120 degrees and radius of 3.

Answer

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

Substitute the given radius in for and the given angle in for to get:

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

Simplify the equation to get the area:

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

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Question

What is the area of the following sector of a full circle?

Note: Figure is not drawn to scale.

Answer

In order to find the fraction of a sector from an angle, you need to know that a full circle is $360^{\circ}$ .

Therefore, we can find the fraction by dividing the angle of the sector by $360^{\circ}$ :

$Percentage = \frac{\measuredangle}{360^{\circ}}$

$Percentage = \frac{60^{\circ}}{360^{\circ}}=\frac{1}{6}$

The formula to find the area of a sector is:

$A=\pi (r^2)(fraction\ of\ whole\ circle)$

where is the radius of the circle.

Plugging in our values, we get:

$A=\pi (r^2)(fraction)$

$A=\pi (9m^2)\left(\frac{1}{6}\right)=13.5\pi m^2$

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Question

Find the area of the shaded region:

Answer

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:

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

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

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Question

Find the area of the following sector:

Answer

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:

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

$A=9 \pi\ m^2$

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Question

In the circle above, the angle A in radians is $\frac{\pi}{4}$

What is the length of arc A?

Answer

Circumference of a Circle = $2\pi r$

Arc Length

$=\frac{Angle A}{2\pi}\ast2\pi r$

$=\frac{\frac{\pi}{4}}{2\pi}\ast2\pi r$

$=\frac{\pi}{(4)2\pi}\ast2\pi r$

$=\frac{1}{8}\ast2\pi r=\frac{\pi r}{4}$

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Question

Figure not drawn to scale.

In the figure above, circle C has a radius of 18, and the measure of angle ACB is equal to 100°. What is the perimeter of the red shaded region?

Answer

The perimeter of any region is the total distance around its boundaries. The perimeter of the shaded region consists of the two straight line segments, AC and BC, as well as the arc AB. In order to find the perimeter of the whole region, we must add the lengths of AC, BC, and the arc AB.

The lengths of AC and BC are both going to be equal to the length of the radius, which is 18. Thus, the perimeter of AC and BC together is 36.

Lastly, we must find the length of arc AB and add it to 36 to get the whole perimeter of the region.

Angle ACB is a central angle, and it intercepts arc AB. The length of AB is going to equal a certain portion of the circumference. This portion will be equal to the ratio of the measure of angle ACB to the measure of the total degrees in the circle. There are 360 degrees in any circle. The ratio of the angle ACB to 360 degrees will be 100/360 = 5/18. Thus, the length of the arc AB will be 5/18 of the circumference of the circle, which equals 2_πr_, according to the formula for circumference.

length of arc AB = (5/18)(2_πr_) = (5/18)(2_π_(18)) = 10_π_.

Thus, the length of arc AB is 10_π_.

The total length of the perimeter is thus 36 + 10_π_.

The answer is 36 + 10_π_.

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Question

If a quarter of the area of a circle is $9\pi$ , then what is a quarter of the circumference of the circle?

Answer

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

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Question

Find the arc length of a sector that has an angle of 120 degrees and radius of 3.

Answer

The equation for the arc length of a sector is $(\frac{x}{360})2\Pi r$ .

Substitute the given radius for and the given angle for to get the following equation:

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

Simplify:

$=(\frac{1}{3})2\Pi (3)$

$=2\Pi$

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Question

$\textup{What is the length, in inches, of an arc with a central angle of }$

$72\textup{ degrees on a circle with a radius of 20 inches?}$

Answer

$\frac{\textup{length of arc}}{\textup{circumference}}=\frac{\textup{central angle}}{360}$

$\textup{Find circumference: }C=2\pi r = 2\pi 20 = 40\pi$

$\frac{x}{40\pi }= \frac{72}{360}::\textup{reduce: }\frac{x}{40\pi }=\frac{1}{5}:\textup{ cross-multiply: }:5x=40\pi$

$x=8\pi$

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Question

Find the circumference of the following sector:

Answer

The formula for the circumference of a sector is

$C = 2 \pi r (part) + 2r$ ,

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

Plugging in our values, we get:

$C = 2 \pi (6m)(\frac{90}{360})+2(6m)$

$C= 3 \pi + 12\ m$

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Question

In the circle above, the length of arc BC is 100 degrees, and the segment AC is a diameter. What is the measure of angle ADB in degrees?

Answer

Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.

Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.

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