QuestionFind the radius and perimeter of sector $OAB$ with area $24\pi \mathrm{cm}^2$ and angle $60^\circ$ . Also, find the area of the circle with $AB$ as diameter.

Studdy Solution

STEP 1

Assumptions1. The area of sector $AB$ is $24\pi$ cm^ . The angle of the sector is $60^\circ$
3. $$ is the centre of the sector4. The area of a sector is given by $\frac{1}{}r^\theta$, where $r$ is the radius and $\theta$ is the angle in radians5. The perimeter of a sector is given by $r + r\theta$, where $r$ is the radius and $\theta$ is the angle in radians6. The area of a circle with diameter $d$ is given by $\frac{1}{4}\pi d^$

STEP 2

First, we need to find the radius of the sector. We can do this by rearranging the formula for the area of a sector to solve for $r$ .
$r = \sqrt{\frac{2 \times \text{Area}}{\theta}}$

STEP 3

Convert the angle from degrees to radians. We know that $180^\circ = \pi$ radians.
$\theta =60^\circ \times \frac{\pi}{180^\circ}$

STEP 4

Calculate the angle in radians.
$\theta =60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \text{ radians}$

STEP 5

Now, plug in the given values for the area and the angle to calculate the radius.
$r = \sqrt{\frac{2 \times24\pi}{\pi/3}}$

STEP 6

Calculate the radius.
$r = \sqrt{\frac{2 \times24\pi \times3}{\pi}} =6 \text{ cm}$

STEP 7

Now that we have the radius, we can find the perimeter of the sector using the formula $2r + r\theta$ .
$\text{Perimeter} =2r + r\theta$

STEP 8

Plug in the values for the radius and the angle to calculate the perimeter.
$\text{Perimeter} =2 \times6 +6 \times \frac{\pi}{3}$

STEP 9

Calculate the perimeter.
$\text{Perimeter} =12 +2\pi =12 +2\pi \text{ cm}$

STEP 10

Now, we need to find the area of the circle with $AB$ as a diameter. The length of $AB$ is $2r$ , so we can use the formula for the area of a circle with diameter $d$ .
$\text{Area} = \frac{}{4}\pi d^2$

STEP 11

Plug in the value for the diameter to calculate the area.
\text{Area} = \frac{}{4}\pi (r)^

STEP 12

Calculate the area.
$\text{Area} = \frac{}{4}\pi (2 \times6)^2 =36\pi \text{ cm}^2$ (a) The radius of the sector is6 cm. (b) The perimeter of the sector is $12 +2\pi$ cm. (c) The area of the circle with $AB$ as a diameter is $36\pi$ cm $^2$ .

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