DSQ: Calculating the area of a sector - GMAT Quantitative

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Question

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: The circle has circumference $40 \pi$ .

Statement 2: $m \angle ACB = 30 ^{\circ }$

Answer

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield the radius, and that can be substituted for in the formula $A = \pi r^{2}$ to find the area. However, it provides no clue that might yield $m \angle AOB$ .

Statement 2 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle that intercepts it - has twice this measure, or $60 ^{\circ }$ . Therefore, Statement 2 alone gives the central angle, but does not yield any clues about the area.

Assume both statements are true. The radius is $40 \pi \div 2 \pi = 20$ and the area is $A = \pi \cdot 20^{2 }= 400 \pi$ . The shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, so the area can be calculated to be $\frac{1}{6} \times 400 \pi = \frac{200 \pi}{3}$ .

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Question

What is the area of a $60^{\circ}$ sector of a circle?

Statement 1: The diameter of the circle is 48 inches.

Statement 2: The length of the arc is $8 \pi$ inches.

Answer

The area of a $60^{\circ}$ sector of radius is

$A= \frac{60}{360} \pi r^{2} = \frac{1}{6} \pi r^{2}$

From the first statement alone, you can halve the diameter to get radius 24 inches.

From the second alone, note that the length of the $60^{\circ}$ arc is

$L= \frac{60}{360} \cdot 2 \pi r = \frac{\pi r}{3}$

Given that length, you can find the radius:

$8 \pi = \frac{\pi r}{3}$

Either way, you can get the radius, so you can calculate the area.

The answer is that either statement alone is sufficient to answer the question.

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Question

The above figure shows two quarter circles inscribed inside a rectangle. What is the total area of the white region?

Statement 1: The area of the black region is $50\pi$ square centimeters.

Statement 2: The rectangle has perimeter 60 centimeters.

Answer

The width of the rectangle is equal to the radius of the quarter circles, which we call ; the length is twice that, or .

The area of the rectangle is $r \cdot 2r = 2r^{2}$ ; the total area of the two black quarter circles is $2 \cdot \frac{1}{4} \cdot \pi r^{2} = \frac{1}{2} \pi r^{2}$ , so the area of the white region is their difference,

$2r^{2} - \frac{1}{2} \pi r^{2}$

Therefore, all that is needed to find the area of the white region is the radius of the quarter circle.

If we know that the area of the black region is $50 \pi$ centimeters, then we can deduce using this equation:

$\frac{1}{2} \pi r^{2} = 50 \pi$

If we know that the perimeter of the rectangle is 60 centimeters, we can deduce via the perimeter formula:

Either statement alone allows us to find the radius and, consequently, the area of the white region.

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Question

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: The sector with central angle $\angle BOD$ has area $48 \pi$ .

Statement 2: $\widehat{BD }= 8 \pi$ .

Answer

Assume Statement 1 alone. No clues are given about the measure of $\angle BOD$ , so that of $\angle AOB$ , and, subsequently, the area of the shaded sector, cannot be determined.

Assume Statement 2 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle $\widehat{BD }$ , or, subsequently, $\widehat{AB}$ , is, and therefore, the central angle of the sector cannot be determined. Also, no information about the area of the circle can be determined.

Now assume both statements are true. Let be the radius of the circle and $N ^{\circ }$ be the measure of $\angle BOD$ . Then:

$\frac{N}{360} \cdot \pi r ^{2} = 48 \pi$

and

$\frac{N}{360} \cdot 2 \pi r = 8 \pi$

The statements can be simplified as

$r ^{2}N = 17,280$

and

From these two statements:

$\frac{r ^{2}N }{rN}=\frac{ 17,280}{1,440}$

; the second statement can be solved for :

.

$m \angle BOD = 120^{\circ }$ , so $m \angle AOB = 60^{\circ }$ .

Since , the circle has area $A = \pi r^{2} = \pi \cdot 12^{2} = 144 \pi$ . Since we know the central angle of the shaded sector as well as the area of the circle, we can calculate the area of the sector as

$\frac{1}{6} \times 144 \pi = 24\pi$ .

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Question

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: $m \angle BCD = 120^{\circ }$ .

Statement 2: The circle has circumference $28 \pi$ .

Answer

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle $\angle BOD$ of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield radius $28 \pi \div 2 \pi = 14$ , and that can be substituted for in the formula $A = \pi r^{2}$ to find the area: $A = \pi \cdot 14^{2} = 196 \pi$ .

However, it provides no clue that might yield $m \angle BOD$ .

From Statement 2 alone, we can find $m \angle BOD$ . $\angle BCD$ , an inscribed angle, intercepts an arc twice its measure - this arc is $\widehat{BAD}$ , which has measure $240^{\circ }$ . $\widehat{B D}$ , the corresponding minor arc, will have measure $360 ^{\circ}- m \widehat{BAD} = 360 ^{\circ} - 240 ^{\circ} = 120 ^{\circ}$ . This gives us $m \angle BOD$ , but no clue that yields the area.

Now assume both statements are true. The area is $196 \pi$ and the shaded sector is $\frac{120^{\circ }}{360^{\circ }}= \frac{1}{3}$ of the circle, so the area can be calculated to be $\frac{1}{3} \times 196 \pi = \frac{196 \pi}{3}$ .

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Question

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1:

Statement 2: $m \angle ACB = 30 ^{\circ }$

Answer

We are asking for the ratio of the areas of the sectors, not the actual areas. The answer is the same regardless of the actual area of the circle, so information about linear measurements such as radius, diameter, and circumference is useless. Statement 2 alone is unhelpful.

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . $\angle ACB$ is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice its measure, or $2 \times 30^{\circ }= 60 ^{\circ }$ . From angle addition, this can be subtracted from $180^{\circ }$ to yield the measure of central angle $\angle BOD$ of the shaded sector, which is $120^{\circ }$ . That makes that sector $\frac{120^{\circ }}{360^{\circ }} = \frac{1}{3}$ of the circle. The white sector is $\frac{2}{3}$ of the circle, and the ratio of the areas can be determined to be $\frac{2}{3} : \frac{1}{3}$ , or .

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Question

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1: $m \angle ACB = 30 ^{\circ }$

Statement 2: $m \angle BOD = 120 ^{\circ }$

Answer

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice this measure, or $60 ^{\circ }$ .

Statement 2 alone asserts that $m \angle BOD = 120 ^{\circ }$ . By angle addition, $m\angle AOB = 180 ^{\circ } - m \angle BOD = 180 ^{\circ } - 120 ^{\circ } = 60 ^{\circ }$ .

Either statement alone tells us that the shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, and that the white sector is $\frac{5}{6}$ of it; it can be subsequently calculated that the ratio of the areas is $\frac{5}{6} : \frac{1}{6}$ , or .

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Question

What is the area of a $60^{\circ}$ sector of a circle?

Statement 1: The diameter of the circle is 48 inches.

Statement 2: The length of the arc is $8 \pi$ inches.

Answer

The area of a $60^{\circ}$ sector of radius is

$A= \frac{60}{360} \pi r^{2} = \frac{1}{6} \pi r^{2}$

From the first statement alone, you can halve the diameter to get radius 24 inches.

From the second alone, note that the length of the $60^{\circ}$ arc is

$L= \frac{60}{360} \cdot 2 \pi r = \frac{\pi r}{3}$

Given that length, you can find the radius:

$8 \pi = \frac{\pi r}{3}$

Either way, you can get the radius, so you can calculate the area.

The answer is that either statement alone is sufficient to answer the question.

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Question

The above figure shows two quarter circles inscribed inside a rectangle. What is the total area of the white region?

Statement 1: The area of the black region is $50\pi$ square centimeters.

Statement 2: The rectangle has perimeter 60 centimeters.

Answer

The width of the rectangle is equal to the radius of the quarter circles, which we call ; the length is twice that, or .

The area of the rectangle is $r \cdot 2r = 2r^{2}$ ; the total area of the two black quarter circles is $2 \cdot \frac{1}{4} \cdot \pi r^{2} = \frac{1}{2} \pi r^{2}$ , so the area of the white region is their difference,

$2r^{2} - \frac{1}{2} \pi r^{2}$

Therefore, all that is needed to find the area of the white region is the radius of the quarter circle.

If we know that the area of the black region is $50 \pi$ centimeters, then we can deduce using this equation:

$\frac{1}{2} \pi r^{2} = 50 \pi$

If we know that the perimeter of the rectangle is 60 centimeters, we can deduce via the perimeter formula:

Either statement alone allows us to find the radius and, consequently, the area of the white region.

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Question

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: The sector with central angle $\angle BOD$ has area $48 \pi$ .

Statement 2: $\widehat{BD }= 8 \pi$ .

Answer

Assume Statement 1 alone. No clues are given about the measure of $\angle BOD$ , so that of $\angle AOB$ , and, subsequently, the area of the shaded sector, cannot be determined.

Assume Statement 2 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle $\widehat{BD }$ , or, subsequently, $\widehat{AB}$ , is, and therefore, the central angle of the sector cannot be determined. Also, no information about the area of the circle can be determined.

Now assume both statements are true. Let be the radius of the circle and $N ^{\circ }$ be the measure of $\angle BOD$ . Then:

$\frac{N}{360} \cdot \pi r ^{2} = 48 \pi$

and

$\frac{N}{360} \cdot 2 \pi r = 8 \pi$

The statements can be simplified as

$r ^{2}N = 17,280$

and

From these two statements:

$\frac{r ^{2}N }{rN}=\frac{ 17,280}{1,440}$

; the second statement can be solved for :

.

$m \angle BOD = 120^{\circ }$ , so $m \angle AOB = 60^{\circ }$ .

Since , the circle has area $A = \pi r^{2} = \pi \cdot 12^{2} = 144 \pi$ . Since we know the central angle of the shaded sector as well as the area of the circle, we can calculate the area of the sector as

$\frac{1}{6} \times 144 \pi = 24\pi$ .

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Question

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: $m \angle BCD = 120^{\circ }$ .

Statement 2: The circle has circumference $28 \pi$ .

Answer

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle $\angle BOD$ of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield radius $28 \pi \div 2 \pi = 14$ , and that can be substituted for in the formula $A = \pi r^{2}$ to find the area: $A = \pi \cdot 14^{2} = 196 \pi$ .

However, it provides no clue that might yield $m \angle BOD$ .

From Statement 2 alone, we can find $m \angle BOD$ . $\angle BCD$ , an inscribed angle, intercepts an arc twice its measure - this arc is $\widehat{BAD}$ , which has measure $240^{\circ }$ . $\widehat{B D}$ , the corresponding minor arc, will have measure $360 ^{\circ}- m \widehat{BAD} = 360 ^{\circ} - 240 ^{\circ} = 120 ^{\circ}$ . This gives us $m \angle BOD$ , but no clue that yields the area.

Now assume both statements are true. The area is $196 \pi$ and the shaded sector is $\frac{120^{\circ }}{360^{\circ }}= \frac{1}{3}$ of the circle, so the area can be calculated to be $\frac{1}{3} \times 196 \pi = \frac{196 \pi}{3}$ .

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Question

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1:

Statement 2: $m \angle ACB = 30 ^{\circ }$

Answer

We are asking for the ratio of the areas of the sectors, not the actual areas. The answer is the same regardless of the actual area of the circle, so information about linear measurements such as radius, diameter, and circumference is useless. Statement 2 alone is unhelpful.

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . $\angle ACB$ is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice its measure, or $2 \times 30^{\circ }= 60 ^{\circ }$ . From angle addition, this can be subtracted from $180^{\circ }$ to yield the measure of central angle $\angle BOD$ of the shaded sector, which is $120^{\circ }$ . That makes that sector $\frac{120^{\circ }}{360^{\circ }} = \frac{1}{3}$ of the circle. The white sector is $\frac{2}{3}$ of the circle, and the ratio of the areas can be determined to be $\frac{2}{3} : \frac{1}{3}$ , or .

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Question

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1: $m \angle ACB = 30 ^{\circ }$

Statement 2: $m \angle BOD = 120 ^{\circ }$

Answer

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice this measure, or $60 ^{\circ }$ .

Statement 2 alone asserts that $m \angle BOD = 120 ^{\circ }$ . By angle addition, $m\angle AOB = 180 ^{\circ } - m \angle BOD = 180 ^{\circ } - 120 ^{\circ } = 60 ^{\circ }$ .

Either statement alone tells us that the shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, and that the white sector is $\frac{5}{6}$ of it; it can be subsequently calculated that the ratio of the areas is $\frac{5}{6} : \frac{1}{6}$ , or .

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Question

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: The circle has circumference $40 \pi$ .

Statement 2: $m \angle ACB = 30 ^{\circ }$

Answer

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield the radius, and that can be substituted for in the formula $A = \pi r^{2}$ to find the area. However, it provides no clue that might yield $m \angle AOB$ .

Statement 2 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle that intercepts it - has twice this measure, or $60 ^{\circ }$ . Therefore, Statement 2 alone gives the central angle, but does not yield any clues about the area.

Assume both statements are true. The radius is $40 \pi \div 2 \pi = 20$ and the area is $A = \pi \cdot 20^{2 }= 400 \pi$ . The shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, so the area can be calculated to be $\frac{1}{6} \times 400 \pi = \frac{200 \pi}{3}$ .

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Question

What is the area of a $60^{\circ}$ sector of a circle?

Statement 1: The diameter of the circle is 48 inches.

Statement 2: The length of the arc is $8 \pi$ inches.

Answer

The area of a $60^{\circ}$ sector of radius is

$A= \frac{60}{360} \pi r^{2} = \frac{1}{6} \pi r^{2}$

From the first statement alone, you can halve the diameter to get radius 24 inches.

From the second alone, note that the length of the $60^{\circ}$ arc is

$L= \frac{60}{360} \cdot 2 \pi r = \frac{\pi r}{3}$

Given that length, you can find the radius:

$8 \pi = \frac{\pi r}{3}$

Either way, you can get the radius, so you can calculate the area.

The answer is that either statement alone is sufficient to answer the question.

Compare your answer with the correct one above

Question

The above figure shows two quarter circles inscribed inside a rectangle. What is the total area of the white region?

Statement 1: The area of the black region is $50\pi$ square centimeters.

Statement 2: The rectangle has perimeter 60 centimeters.

Answer

The width of the rectangle is equal to the radius of the quarter circles, which we call ; the length is twice that, or .

The area of the rectangle is $r \cdot 2r = 2r^{2}$ ; the total area of the two black quarter circles is $2 \cdot \frac{1}{4} \cdot \pi r^{2} = \frac{1}{2} \pi r^{2}$ , so the area of the white region is their difference,

$2r^{2} - \frac{1}{2} \pi r^{2}$

Therefore, all that is needed to find the area of the white region is the radius of the quarter circle.

If we know that the area of the black region is $50 \pi$ centimeters, then we can deduce using this equation:

$\frac{1}{2} \pi r^{2} = 50 \pi$

If we know that the perimeter of the rectangle is 60 centimeters, we can deduce via the perimeter formula:

Either statement alone allows us to find the radius and, consequently, the area of the white region.

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Question

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: The sector with central angle $\angle BOD$ has area $48 \pi$ .

Statement 2: $\widehat{BD }= 8 \pi$ .

Answer

Assume Statement 1 alone. No clues are given about the measure of $\angle BOD$ , so that of $\angle AOB$ , and, subsequently, the area of the shaded sector, cannot be determined.

Assume Statement 2 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle $\widehat{BD }$ , or, subsequently, $\widehat{AB}$ , is, and therefore, the central angle of the sector cannot be determined. Also, no information about the area of the circle can be determined.

Now assume both statements are true. Let be the radius of the circle and $N ^{\circ }$ be the measure of $\angle BOD$ . Then:

$\frac{N}{360} \cdot \pi r ^{2} = 48 \pi$

and

$\frac{N}{360} \cdot 2 \pi r = 8 \pi$

The statements can be simplified as

$r ^{2}N = 17,280$

and

From these two statements:

$\frac{r ^{2}N }{rN}=\frac{ 17,280}{1,440}$

; the second statement can be solved for :

.

$m \angle BOD = 120^{\circ }$ , so $m \angle AOB = 60^{\circ }$ .

Since , the circle has area $A = \pi r^{2} = \pi \cdot 12^{2} = 144 \pi$ . Since we know the central angle of the shaded sector as well as the area of the circle, we can calculate the area of the sector as

$\frac{1}{6} \times 144 \pi = 24\pi$ .

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Question

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: $m \angle BCD = 120^{\circ }$ .

Statement 2: The circle has circumference $28 \pi$ .

Answer

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle $\angle BOD$ of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield radius $28 \pi \div 2 \pi = 14$ , and that can be substituted for in the formula $A = \pi r^{2}$ to find the area: $A = \pi \cdot 14^{2} = 196 \pi$ .

However, it provides no clue that might yield $m \angle BOD$ .

From Statement 2 alone, we can find $m \angle BOD$ . $\angle BCD$ , an inscribed angle, intercepts an arc twice its measure - this arc is $\widehat{BAD}$ , which has measure $240^{\circ }$ . $\widehat{B D}$ , the corresponding minor arc, will have measure $360 ^{\circ}- m \widehat{BAD} = 360 ^{\circ} - 240 ^{\circ} = 120 ^{\circ}$ . This gives us $m \angle BOD$ , but no clue that yields the area.

Now assume both statements are true. The area is $196 \pi$ and the shaded sector is $\frac{120^{\circ }}{360^{\circ }}= \frac{1}{3}$ of the circle, so the area can be calculated to be $\frac{1}{3} \times 196 \pi = \frac{196 \pi}{3}$ .

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Question

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1:

Statement 2: $m \angle ACB = 30 ^{\circ }$

Answer

We are asking for the ratio of the areas of the sectors, not the actual areas. The answer is the same regardless of the actual area of the circle, so information about linear measurements such as radius, diameter, and circumference is useless. Statement 2 alone is unhelpful.

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . $\angle ACB$ is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice its measure, or $2 \times 30^{\circ }= 60 ^{\circ }$ . From angle addition, this can be subtracted from $180^{\circ }$ to yield the measure of central angle $\angle BOD$ of the shaded sector, which is $120^{\circ }$ . That makes that sector $\frac{120^{\circ }}{360^{\circ }} = \frac{1}{3}$ of the circle. The white sector is $\frac{2}{3}$ of the circle, and the ratio of the areas can be determined to be $\frac{2}{3} : \frac{1}{3}$ , or .

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Question

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1: $m \angle ACB = 30 ^{\circ }$

Statement 2: $m \angle BOD = 120 ^{\circ }$

Answer

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice this measure, or $60 ^{\circ }$ .

Statement 2 alone asserts that $m \angle BOD = 120 ^{\circ }$ . By angle addition, $m\angle AOB = 180 ^{\circ } - m \angle BOD = 180 ^{\circ } - 120 ^{\circ } = 60 ^{\circ }$ .

Either statement alone tells us that the shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, and that the white sector is $\frac{5}{6}$ of it; it can be subsequently calculated that the ratio of the areas is $\frac{5}{6} : \frac{1}{6}$ , or .

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