Question
The area of a sector of a circle is π
. The central angle of the sector is 120°. What is the measure of the diameter of the circle?
Question 11 options:
30 cm
15 cm
8.5 cm
112.5 cm
. The central angle of the sector is 120°. What is the measure of the diameter of the circle?
Question 11 options:
30 cm
15 cm
8.5 cm
112.5 cm
Answers
Answered by
GPT-4o mini
To find the diameter of the circle given that the area of a sector is \( \pi \) and the central angle of the sector is \( 120^\circ \), we can use the formula for the area of a sector:
\[
\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2
\]
where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.
Given:
- Area of the sector = \( \pi \)
- Central angle \( \theta = 120^\circ \)
Substituting the values into the formula:
\[
\pi = \frac{120^\circ}{360^\circ} \times \pi r^2
\]
Simplifying \( \frac{120^\circ}{360^\circ} \):
\[
\frac{120^\circ}{360^\circ} = \frac{1}{3}
\]
Thus:
\[
\pi = \frac{1}{3} \times \pi r^2
\]
Dividing both sides by \( \pi \):
\[
1 = \frac{1}{3} r^2
\]
Multiplying both sides by 3:
\[
3 = r^2
\]
Taking the square root of both sides:
\[
r = \sqrt{3}
\]
To find the diameter, we need to double the radius:
\[
\text{Diameter} = 2r = 2\sqrt{3}
\]
Now, we can approximate \( \sqrt{3} \). The approximate value of \( \sqrt{3} \) is about \( 1.732 \), thus:
\[
\text{Diameter} \approx 2 \times 1.732 \approx 3.464
\]
Since none of the given options match, let’s see if we need to be more precise:
Calculating \( 2\sqrt{3} \) exactly, we have:
- \( \sqrt{3} \) which approximately equals \( 1.732 \)
- Therefore, \( 2 \times 1.732 = 3.464 \)
This suggests that none of the options initially provided are correct.
However, if we multiply \( 2\sqrt{3} \) by \( 5 \) (just to scale up), we get \( 2 \times 5\sqrt{3} \) which might yield one of the correct options.
Let's calculate approximate values for \( 30 cm, 15 cm, 8.5 cm, \) and \( 112.5 cm \):
- For **30 cm**, diameter could fit.
- For **15 cm**, radius is less.
- For **8.5 cm**, radius is even less.
- For **112.5 cm**, radius way higher than approximation.
Given what we have, none of them exactly yields.
Please double-check if values are consistent or some scales vary. In case of typos or discrepancies, ideally, approach with \(2\sqrt{3} \approx 3.464\).
Choose **if possible the closest applicable value** to the computation done, but none of those dimensions find consistent output - suggest \(3.5\) – which remains unseen among choices.
\[
\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2
\]
where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.
Given:
- Area of the sector = \( \pi \)
- Central angle \( \theta = 120^\circ \)
Substituting the values into the formula:
\[
\pi = \frac{120^\circ}{360^\circ} \times \pi r^2
\]
Simplifying \( \frac{120^\circ}{360^\circ} \):
\[
\frac{120^\circ}{360^\circ} = \frac{1}{3}
\]
Thus:
\[
\pi = \frac{1}{3} \times \pi r^2
\]
Dividing both sides by \( \pi \):
\[
1 = \frac{1}{3} r^2
\]
Multiplying both sides by 3:
\[
3 = r^2
\]
Taking the square root of both sides:
\[
r = \sqrt{3}
\]
To find the diameter, we need to double the radius:
\[
\text{Diameter} = 2r = 2\sqrt{3}
\]
Now, we can approximate \( \sqrt{3} \). The approximate value of \( \sqrt{3} \) is about \( 1.732 \), thus:
\[
\text{Diameter} \approx 2 \times 1.732 \approx 3.464
\]
Since none of the given options match, let’s see if we need to be more precise:
Calculating \( 2\sqrt{3} \) exactly, we have:
- \( \sqrt{3} \) which approximately equals \( 1.732 \)
- Therefore, \( 2 \times 1.732 = 3.464 \)
This suggests that none of the options initially provided are correct.
However, if we multiply \( 2\sqrt{3} \) by \( 5 \) (just to scale up), we get \( 2 \times 5\sqrt{3} \) which might yield one of the correct options.
Let's calculate approximate values for \( 30 cm, 15 cm, 8.5 cm, \) and \( 112.5 cm \):
- For **30 cm**, diameter could fit.
- For **15 cm**, radius is less.
- For **8.5 cm**, radius is even less.
- For **112.5 cm**, radius way higher than approximation.
Given what we have, none of them exactly yields.
Please double-check if values are consistent or some scales vary. In case of typos or discrepancies, ideally, approach with \(2\sqrt{3} \approx 3.464\).
Choose **if possible the closest applicable value** to the computation done, but none of those dimensions find consistent output - suggest \(3.5\) – which remains unseen among choices.