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

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Question

Circle A has twice the radius of Circle B. Which is the greater quantity?

(a) The area of a $90^{\circ }$ sector of Circle A

(b) The area of Circle B

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Answer

Let be the radius of Circle B. The radius of Circle A is therefore .

A $90^{\circ }$ sector of a circle comprises $\frac{90}{360} = \frac{1}{4}$ of the circle. The $90^{\circ }$ sector of circle A has area $\frac{1}{4} \pi (2r)^{2} = \frac{1}{4} \cdot 4 \pi r^{2} = \pi r^{2}$ , the area of Circle B. The two quantities are equal.

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Question

What is the area, rounded to the nearest hundredth, of the sector shaded in circle O in the diagram above?

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Answer

To find the area of a sector, you need to find a percentage of the total area of the circle. You do this by dividing the sector angle by the total number of degrees in a full circle (i.e. ˚). Thus, for our circle, which has a sector with an angle of ˚, we have a percentage of:

$\frac{81}{360}$

Now, we will multiply this by the total area of the circle. Recall that we find such an area according to the equation:

$A = \pi * r^2$

For our problem,

Therefore, our equation is:

$\frac{81}{360} * \pi * 12.3^2 = \frac{12254.49\pi}{360}$

Using your calculator, you can determine that this is approximately .

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Question

What is the area, rounded to the nearest hundredth, of the sector shaded in circle O in the diagram above?

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Answer

To find the area of a sector, you need to find a percentage of the total area of the circle. You do this by dividing the sector angle by the total number of degrees in a full circle (i.e. ˚). Thus, for our circle, which has a sector with an angle of ˚, we have a percentage of:

$\frac{122}{360}$

Now, we will multiply this by the total area of the circle. Recall that we find such an area according to the equation:

$A = \pi * r^2$

For our problem,

Therefore, our equation is:

$\frac{122}{360} * \pi * 9.12^2 = \frac{10147.2768\pi}{360}$

Using your calculator, you can determine that this is approximately .

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Question

Refer to the above figure, Which is the greater quantity?

(a) The area of the orange semicircle

(b) The area of $\bigtriangleup ABC$

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Answer

$\bigtriangleup ABC$ has two angles of degree measure 60; its third angle must also have measure 60, making $\bigtriangleup ABC$ an equilateral triangle

For the sake of simplicity, let ; the reasoning is independent of the actual length. The area of equilateral $\bigtriangleup ABC$ can be found by substituting in the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

$= \frac{1^{2}\sqrt{3}}{4}$

$= \frac{1 \cdot \sqrt{3}}{4}$

$\approx \frac{1.7}{4}$

$\approx 0.425$

Also, if , then the orange semicircle has diameter 1 and radius $\frac{1}{2}$ . Its area can be found by substituting $r = \frac{1}{2}$ in the formula:

$A = \frac{1}{2} \cdot \pi r^{2}$

$= \frac{1}{2} \cdot \pi \left ( \frac{1}{2} \right ) ^{2}$

$= \frac{1}{2} \cdot \pi \cdot \left ( \frac{1}{4 } \right )$

$= \frac{1}{8} \cdot \pi$

$\approx 0. 125 \cdot 3.14$

$\approx 0. 3925$

$\bigtriangleup ABC$ has a greater area than the orange semicircle.

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Question

Refer to the above figure, Which is the greater quantity?

(a) The area of $\bigtriangleup ABC$

(b) The area of the orange semicircle

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Answer

$\bigtriangleup ABC$ has two angles of degree measure 45; the third angle must measure 90 degrees, making $\bigtriangleup ABC$ a right triangle.

For the sake of simplicity, let ; the reasoning is independent of the actual length. The legs of a 45-45-90 triangle are congruent, so . The area of a right triangle is half the product of its legs, so

$A = \frac{1}{2} \cdot 1 \cdot 1 = \frac{1}{2} = 0.5$

Also, if , then the orange semicircle has diameter 1 and radius $\frac{1}{2}$ . Its area can be found by substituting $r = \frac{1}{2}$ in the formula:

$A = \frac{1}{2} \cdot \pi r^{2}$

$= \frac{1}{2} \cdot \pi \left ( \frac{1}{2} \right ) ^{2}$

$= \frac{1}{2} \cdot \pi \cdot \left ( \frac{1}{4 } \right )$

$= \frac{1}{8} \cdot \pi$

$\approx 0. 125 \cdot 3.14$

$\approx 0. 3925$

$\bigtriangleup ABC$ has a greater area than the orange semicircle.

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Question

The above circle, which is divided into sectors of equal size, has diameter 20. Give the area of the shaded region.

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Answer

The radius of a circle is half its diameter; the radius of the circle in the diagram is half of 20, or 10.

To find the area of the circle, set in the area formula:

$A = \pi r ^{2} = \pi \cdot 10 ^{2} = 100 \pi$

The circle is divided into sixteen sectors of equal size, five of which are shaded; the shaded portion is

$\frac{5}{16} \cdot 100 \pi = \frac{500}{16} \pi = \frac{125 \pi }{4 }$ .

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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 $65^{\circ}$ . 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 $65^{\circ}$ . 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.

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

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

Circle A has twice the radius of Circle B. Which is the greater quantity?

(a) The area of a $90^{\circ }$ sector of Circle A

(b) The area of Circle B

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Answer

Let be the radius of Circle B. The radius of Circle A is therefore .

A $90^{\circ }$ sector of a circle comprises $\frac{90}{360} = \frac{1}{4}$ of the circle. The $90^{\circ }$ sector of circle A has area $\frac{1}{4} \pi (2r)^{2} = \frac{1}{4} \cdot 4 \pi r^{2} = \pi r^{2}$ , the area of Circle B. The two quantities are equal.

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Question

What is the area, rounded to the nearest hundredth, of the sector shaded in circle O in the diagram above?

Tap to reveal answer

Answer

To find the area of a sector, you need to find a percentage of the total area of the circle. You do this by dividing the sector angle by the total number of degrees in a full circle (i.e. ˚). Thus, for our circle, which has a sector with an angle of ˚, we have a percentage of:

$\frac{81}{360}$

Now, we will multiply this by the total area of the circle. Recall that we find such an area according to the equation:

$A = \pi * r^2$

For our problem,

Therefore, our equation is:

$\frac{81}{360} * \pi * 12.3^2 = \frac{12254.49\pi}{360}$

Using your calculator, you can determine that this is approximately .

← Didn't Know|Knew It →

Question

What is the area, rounded to the nearest hundredth, of the sector shaded in circle O in the diagram above?

Tap to reveal answer

Answer

To find the area of a sector, you need to find a percentage of the total area of the circle. You do this by dividing the sector angle by the total number of degrees in a full circle (i.e. ˚). Thus, for our circle, which has a sector with an angle of ˚, we have a percentage of:

$\frac{122}{360}$

Now, we will multiply this by the total area of the circle. Recall that we find such an area according to the equation:

$A = \pi * r^2$

For our problem,

Therefore, our equation is:

$\frac{122}{360} * \pi * 9.12^2 = \frac{10147.2768\pi}{360}$

Using your calculator, you can determine that this is approximately .

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Question

Refer to the above figure, Which is the greater quantity?

(a) The area of the orange semicircle

(b) The area of $\bigtriangleup ABC$

Tap to reveal answer

Answer

$\bigtriangleup ABC$ has two angles of degree measure 60; its third angle must also have measure 60, making $\bigtriangleup ABC$ an equilateral triangle

For the sake of simplicity, let ; the reasoning is independent of the actual length. The area of equilateral $\bigtriangleup ABC$ can be found by substituting in the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

$= \frac{1^{2}\sqrt{3}}{4}$

$= \frac{1 \cdot \sqrt{3}}{4}$

$\approx \frac{1.7}{4}$

$\approx 0.425$

Also, if , then the orange semicircle has diameter 1 and radius $\frac{1}{2}$ . Its area can be found by substituting $r = \frac{1}{2}$ in the formula:

$A = \frac{1}{2} \cdot \pi r^{2}$

$= \frac{1}{2} \cdot \pi \left ( \frac{1}{2} \right ) ^{2}$

$= \frac{1}{2} \cdot \pi \cdot \left ( \frac{1}{4 } \right )$

$= \frac{1}{8} \cdot \pi$

$\approx 0. 125 \cdot 3.14$

$\approx 0. 3925$

$\bigtriangleup ABC$ has a greater area than the orange semicircle.

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Question

Refer to the above figure, Which is the greater quantity?

(a) The area of $\bigtriangleup ABC$

(b) The area of the orange semicircle

Tap to reveal answer

Answer

$\bigtriangleup ABC$ has two angles of degree measure 45; the third angle must measure 90 degrees, making $\bigtriangleup ABC$ a right triangle.

For the sake of simplicity, let ; the reasoning is independent of the actual length. The legs of a 45-45-90 triangle are congruent, so . The area of a right triangle is half the product of its legs, so

$A = \frac{1}{2} \cdot 1 \cdot 1 = \frac{1}{2} = 0.5$

Also, if , then the orange semicircle has diameter 1 and radius $\frac{1}{2}$ . Its area can be found by substituting $r = \frac{1}{2}$ in the formula:

$A = \frac{1}{2} \cdot \pi r^{2}$

$= \frac{1}{2} \cdot \pi \left ( \frac{1}{2} \right ) ^{2}$

$= \frac{1}{2} \cdot \pi \cdot \left ( \frac{1}{4 } \right )$

$= \frac{1}{8} \cdot \pi$

$\approx 0. 125 \cdot 3.14$

$\approx 0. 3925$

$\bigtriangleup ABC$ has a greater area than the orange semicircle.

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Question

The above circle, which is divided into sectors of equal size, has diameter 20. Give the area of the shaded region.

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Answer

The radius of a circle is half its diameter; the radius of the circle in the diagram is half of 20, or 10.

To find the area of the circle, set in the area formula:

$A = \pi r ^{2} = \pi \cdot 10 ^{2} = 100 \pi$

The circle is divided into sixteen sectors of equal size, five of which are shaded; the shaded portion is

$\frac{5}{16} \cdot 100 \pi = \frac{500}{16} \pi = \frac{125 \pi }{4 }$ .

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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 $65^{\circ}$ . 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 $65^{\circ}$ . 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.

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

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

Circle A has twice the radius of Circle B. Which is the greater quantity?

(a) The area of a $90^{\circ }$ sector of Circle A

(b) The area of Circle B

Tap to reveal answer

Answer

Let be the radius of Circle B. The radius of Circle A is therefore .

A $90^{\circ }$ sector of a circle comprises $\frac{90}{360} = \frac{1}{4}$ of the circle. The $90^{\circ }$ sector of circle A has area $\frac{1}{4} \pi (2r)^{2} = \frac{1}{4} \cdot 4 \pi r^{2} = \pi r^{2}$ , the area of Circle B. The two quantities are equal.

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Question

What is the area, rounded to the nearest hundredth, of the sector shaded in circle O in the diagram above?

Tap to reveal answer

Answer

To find the area of a sector, you need to find a percentage of the total area of the circle. You do this by dividing the sector angle by the total number of degrees in a full circle (i.e. ˚). Thus, for our circle, which has a sector with an angle of ˚, we have a percentage of:

$\frac{81}{360}$

Now, we will multiply this by the total area of the circle. Recall that we find such an area according to the equation:

$A = \pi * r^2$

For our problem,

Therefore, our equation is:

$\frac{81}{360} * \pi * 12.3^2 = \frac{12254.49\pi}{360}$

Using your calculator, you can determine that this is approximately .

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