Sectors

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PSAT Math › Sectors

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Sector

Note: Figure NOT drawn to scale.

In the above circle, the length of arc $\widehat{MN}$ is $8 \pi$ , and the length of arc $\widehat{MYN}$ is $42 \pi$ . Evaluate .

$57 \frac{3}{5}^{\circ}$

$68\frac{4}{7} ^{\circ }$

$72^{\circ }$

$66\frac{2}{3}^{ \circ}$

$60 ^{\circ}$

Explanation

The circumference of the circle is the sum of the lengths of the arcs $\widehat{MN}$ and $\widehat{MYN}$ , which is

$8 \pi + 42 \pi = 50 \pi$

$\widehat{MN}$ is therefore

$\frac{8 \pi}{50 \pi } = \frac{4}{25}$

of the circle, and its degree measure is

$\frac{4}{25} \cdot 360 ^{\circ} = 57 \frac{3}{5}^{\circ}$

Sector

Note: Figure NOT drawn to scale.

Refer to the above figure. The ratio of the area of the white sector to that of the gray sector is 5 to 1. Evaluate .

$60^{\circ }$

$72^{\circ }$

$80^{\circ }$

$66^{\circ }$

$54^{\circ }$

Explanation

The ratio of the areas is 5 to 1, so the white sector is one sixth of the circle. This means that the central angle of the white sector is one sixth of $\frac{1}{6} \cdot 360^{\circ} = 60^{\circ }$ .

Sector

Note: Figure NOT drawn to scale.

In the above circle, the length of arc $\widehat{MN}$ is $8 \pi$ , and the length of arc $\widehat{MYN}$ is $42 \pi$ . Evaluate .

$57 \frac{3}{5}^{\circ}$

$68\frac{4}{7} ^{\circ }$

$72^{\circ }$

$66\frac{2}{3}^{ \circ}$

$60 ^{\circ}$

Explanation

The circumference of the circle is the sum of the lengths of the arcs $\widehat{MN}$ and $\widehat{MYN}$ , which is

$8 \pi + 42 \pi = 50 \pi$

$\widehat{MN}$ is therefore

$\frac{8 \pi}{50 \pi } = \frac{4}{25}$

of the circle, and its degree measure is

$\frac{4}{25} \cdot 360 ^{\circ} = 57 \frac{3}{5}^{\circ}$

Sector

Note: Figure NOT drawn to scale.

Refer to the above figure. The ratio of the area of the white sector to that of the gray sector is 5 to 1. Evaluate .

$60^{\circ }$

$72^{\circ }$

$80^{\circ }$

$66^{\circ }$

$54^{\circ }$

Explanation

The ratio of the areas is 5 to 1, so the white sector is one sixth of the circle. This means that the central angle of the white sector is one sixth of $\frac{1}{6} \cdot 360^{\circ} = 60^{\circ }$ .

Sector

Note: Figure NOT drawn to scale.

The area of the gray sector in the above circle is $6 \pi$ . The area of the white sector is $42 \pi$ . Evaluate .

$45^{\circ }$

$60^{\circ }$

$80^{\circ }$

$51 \frac{3}{7} ^{\circ }$

$66\frac{2}{3} ^{\circ }$

Explanation

The total area of the circle is the sum of the areas of the white and gray sectors, or

$6 \pi + 42 \pi = 48 \pi$

The gray sector takes up

$\frac{6 \pi}{48 \pi} = \frac{1}{8}$

of the circle, so the degree measure of the gray sector is equal to

$\frac{1}{8} \times 360 ^{\circ} = 45 ^{\circ}$

Sector

Note: Figure NOT drawn to scale.

The area of the gray sector in the above circle is $6 \pi$ . The area of the white sector is $42 \pi$ . Evaluate .

$45^{\circ }$

$60^{\circ }$

$80^{\circ }$

$51 \frac{3}{7} ^{\circ }$

$66\frac{2}{3} ^{\circ }$

Explanation

The total area of the circle is the sum of the areas of the white and gray sectors, or

$6 \pi + 42 \pi = 48 \pi$

The gray sector takes up

$\frac{6 \pi}{48 \pi} = \frac{1}{8}$

of the circle, so the degree measure of the gray sector is equal to

$\frac{1}{8} \times 360 ^{\circ} = 45 ^{\circ}$

What is the angle of a sector of area on a circle having a radius of ?

$22.92^{\circ}$

$0.06^{\circ}$

$15.22^{\circ}$

$3.00^{\circ}$

$72.00^{\circ}$

Explanation

To begin, you should compute the complete area of the circle:

$A=\pi r^2$

For your data, this is:

$A=15^2\pi=225\pi$

Now, to find the angle measure of a sector, you find what portion of the circle the sector is. Here, it is:

$\frac{45}{225\pi}$

Now, multiply this by the total degrees in a circle:

$\frac{45}{225\pi}*360=22.918311805212$

Rounded, this is $22.92^{\circ}$ .

What is the angle of a sector of area on a circle having a radius of ?

$22.92^{\circ}$

$0.06^{\circ}$

$15.22^{\circ}$

$3.00^{\circ}$

$72.00^{\circ}$

Explanation

To begin, you should compute the complete area of the circle:

$A=\pi r^2$

For your data, this is:

$A=15^2\pi=225\pi$

Now, to find the angle measure of a sector, you find what portion of the circle the sector is. Here, it is:

$\frac{45}{225\pi}$

Now, multiply this by the total degrees in a circle:

$\frac{45}{225\pi}*360=22.918311805212$

Rounded, this is $22.92^{\circ}$ .

Sector

Note: Figure NOT drawn to scale.

In the above circle, . Give the ratio of the area of the white sector to that of the gray sector.

Explanation

A $72^{\circ}$ sector is $\frac{72^{\circ }}{360^{\circ}}= \frac{1}{5}$ of the circle. The white sector is therefore $\frac{4}{5}$ of the circle, and the ratio of their areas is

$\frac{4}{5} : \frac{1}{5}$ ,

which simplifies to

Sector

Note: Figure NOT drawn to scale.

In the above circle, . Give the ratio of the area of the white sector to that of the gray sector.

Explanation

A $72^{\circ}$ sector is $\frac{72^{\circ }}{360^{\circ}}= \frac{1}{5}$ of the circle. The white sector is therefore $\frac{4}{5}$ of the circle, and the ratio of their areas is

$\frac{4}{5} : \frac{1}{5}$ ,

which simplifies to

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