Chandler Ortega
08/05/2024 · Elementary School
Find the central angle \( \theta \) which forms a sector of area 3 square meters of a circle of radius 6 meters. \( \theta \approx \square \) radian (Type an integer or decimal rounded to three decimal places as needed.)
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To find the central angle \( \theta \) that forms a sector of area 3 square meters in a circle of radius 6 meters, we can use the formula for the area of a sector:
\[
A = \frac{1}{2} r^2 \theta
\]
where \( A \) is the area of the sector, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians.
Given:
- \( A = 3 \) square meters
- \( r = 6 \) meters
We can substitute these values into the formula:
\[
3 = \frac{1}{2} (6^2) \theta
\]
Calculating \( 6^2 \):
\[
6^2 = 36
\]
Now substituting this back into the equation:
\[
3 = \frac{1}{2} \cdot 36 \cdot \theta
\]
This simplifies to:
\[
3 = 18 \theta
\]
To solve for \( \theta \), we divide both sides by 18:
\[
\theta = \frac{3}{18} = \frac{1}{6}
\]
Now, we can convert \( \frac{1}{6} \) to a decimal:
\[
\theta \approx 0.1667
\]
Rounding to three decimal places, we have:
\[
\theta \approx 0.167
\]
Thus, the central angle \( \theta \) is approximately:
\[
\boxed{0.167} \text{ radians}
\]
Quick Answer
The central angle \( \theta \) is approximately 0.167 radians.
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