Question Find the area of the sector with central angle $76^{\circ}$ in a circle with radius $11$ . Round to the nearest hundredth.

Studdy Solution

STEP 1

Assumptions
1. Circle N has a central angle $m \angle MNP$ which is $76^{\circ}$ .
2. The radius $MN$ of circle N is 11 units.
3. The area of sector MNP is to be found.
4. The area of the sector is proportional to the central angle of the sector.
5. The formula for the area of a sector with central angle $\theta$ in degrees and radius $r$ is given by: $Area\ of\ sector = \frac{\theta}{360} \times \pi r^2$

STEP 2

First, we need to calculate the area of the entire circle using the formula for the area of a circle, which is $\pi r^2$ .
$Area\ of\ circle = \pi r^2$

STEP 3

Now, plug in the given value for the radius $r$ to calculate the area of the circle.
$Area\ of\ circle = \pi (11)^2$

STEP 4

Calculate the area of the entire circle.
$Area\ of\ circle = \pi (121)$

STEP 5

Next, we calculate the area of sector MNP using the formula for the area of a sector.
$Area\ of\ sector\ MNP = \frac{m \angle MNP}{360} \times \pi r^2$

STEP 6

Plug in the given values for the central angle $m \angle MNP$ and the radius $r$ to calculate the area of sector MNP.
$Area\ of\ sector\ MNP = \frac{76}{360} \times \pi (11)^2$

STEP 7

Calculate the area of sector MNP.
$Area\ of\ sector\ MNP = \frac{76}{360} \times \pi (121)$

STEP 8

Simplify the fraction $\frac{76}{360}$ .
$Area\ of\ sector\ MNP = \frac{19}{90} \times \pi (121)$

STEP 9

Calculate the numerical value of the area of sector MNP.
$Area\ of\ sector\ MNP = \frac{19}{90} \times \pi \times 121$

STEP 10

Use a calculator to find the numerical value of the area of sector MNP to the nearest hundredth.
$Area\ of\ sector\ MNP \approx \frac{19}{90} \times 3.14159 \times 121$

STEP 11

Perform the multiplication to find the area of sector MNP.
$Area\ of\ sector\ MNP \approx \frac{19}{90} \times 3.14159 \times 121 \approx 80.42$
The area of sector MNP is approximately $80.42$ square units.

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