How to find the area of a sector - ISEE Upper Level Quantitative Reasoning

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Question

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

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Answer

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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Question

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

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Answer

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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Question

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

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Answer

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . 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.

$A_${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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Question

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

Tap to reveal answer

Answer

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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Question

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

Tap to reveal answer

Answer

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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Question

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

Tap to reveal answer

Answer

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . 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.

$A_${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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Question

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

Tap to reveal answer

Answer

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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Question

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

Tap to reveal answer

Answer

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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Question

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

Tap to reveal answer

Answer

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . 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.

$A_${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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Question

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

Tap to reveal answer

Answer

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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Question

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

Tap to reveal answer

Answer

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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Question

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

Tap to reveal answer

Answer

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . 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.

$A_${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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