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ISEE Upper Level Math : Sectors

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Example Question #1 : How To Find The Angle For A Percentage Of A Circle

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

Possible Answers:

$15.5^{\circ}$

$430^{\circ}$

$154.8^{\circ}$

$366^{\circ}$

Correct answer:

$154.8^{\circ}$

Explanation:

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

Use the following formula and solve for x:

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

Begin by dividing over the 100

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

Then multiply by 360

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

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Example Question #1 : How To Find The Angle For A Percentage Of A Circle

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

Possible Answers:

$262^{\circ}$

$145^{\circ}$

$45^{\circ}$

$162^{\circ}$

Correct answer:

$162^{\circ}$

Explanation:

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.

$45\%\rightarrow 0.45$

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

So, our answer is 162 degrees.

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Example Question #1 : How To Find The Area Of A Sector

Sector

Give the area of the white region of the above circle if $\overarc{AB}$ has length $12 \pi$ .

Possible Answers:

$567 \pi$

$81 \pi$

$729 \pi$

$243 \pi$

Correct answer:

$567 \pi$

Explanation:

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

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

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

$C =54\pi$

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

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

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

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

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

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

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Example Question #2 : How To Find The Area Of A Sector

Sector

The circumference of the above circle is $30 \pi$ . Give the area of the shaded region.

Possible Answers:

$50 \pi$

$400 \pi$

$100 \pi$

$25 \pi$

Correct answer:

$50 \pi$

Explanation:

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

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

The area of the entire circle can be found by substituting for in the formula:

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

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

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

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Example Question #1 : How To Find The Area Of A Sector

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

Possible Answers:

2.89 in^2

$9.08\pi in^2$

$14.23\pi in^2$

$2.89\pi in^2$

Correct answer:

$2.89\pi in^2$

Explanation:

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

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

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

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

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

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Example Question #1 : Sectors

A giant clock has a minute hand six feet long. How far, in inches, did the tip move between noon and 1:20 PM?

Possible Answers:

$768 \pi \textrm{ in}$

$48 \pi \textrm{ in}$

$96\pi \textrm{ in}$

$384 \pi \textrm{ in}$

$192 \pi \textrm{ in}$

Correct answer:

$192 \pi \textrm{ in}$

Explanation:

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 $2\pi r = 2 \pi \times 6 = 12\pi$ feet. One hour and twenty minutes is $1 \frac{1}{3}$ hours, so the tip of the hand moved $1 \frac{1}{3} \times 12 \pi = 16 \pi$ feet, or $16 \pi \times 12 = 192 \pi$ inches.

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Example Question #2 : Sectors

A giant clock has a minute hand three feet long. How far, in inches, did the tip move between noon and 12:20 PM?

Possible Answers:

$36\pi \textrm{ in}$

$12\pi \textrm{ in}$

It is impossible to tell from the information given

$24\pi \textrm{ in}$

$18\pi \textrm{ in}$

Correct answer:

$24\pi \textrm{ in}$

Explanation:

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

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Example Question #3 : Sectors

Inscribed

In the above figure, express in terms of .

Possible Answers:

$y = \frac{1}{2}x - 10$

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

$y = x- \frac{1}{4}$

$y = \frac{1}{2}x - \frac{3}{2}$

Correct answer:

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

Explanation:

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

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

4y + 13 = 2 (2x- 7)

4y + 13 = 4x- 14

4y + 13 - 13 = 4x- 14 - 13

4y = 4x- 27

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

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

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Example Question #4 : Sectors

Intercepted

In the above diagram, radius OA = 12 .

Give the length of $\overarc {AB}$ .

Possible Answers:

$\frac{288\pi }{5}$

$\frac{24 \pi }{5}$

$\frac{72 \pi }{5}$

$\frac{48 \pi }{5}$

Correct answer:

$\frac{48 \pi }{5}$

Explanation:

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

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

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

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

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

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

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Example Question #5 : Sectors

Possible Answers:

1.44in

$4.54\pi in$

4.54in

$3.01\pi in$

Correct answer:

4.54in

Explanation:

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

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

Now, just plug in and simplify

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

So, our answer is 4.54in

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ISEE Upper Level Math : Sectors

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : How To Find The Angle For A Percentage Of A Circle

Example Question #1 : How To Find The Angle For A Percentage Of A Circle

Example Question #1 : How To Find The Area Of A Sector

Example Question #2 : How To Find The Area Of A Sector

Example Question #1 : How To Find The Area Of A Sector

Example Question #1 : Sectors

Example Question #2 : Sectors

Example Question #3 : Sectors

Example Question #4 : Sectors

Example Question #5 : Sectors

All ISEE Upper Level Math Resources

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