Find the area of the circle that contains the sector described below. Round to the nearest tenth, if necessary.
One sector has an area of 210 square centimeters and an arc measure of 30°.
210 * 360/30
To find the area of the circle that contains the given sector, we need to first find the radius of the circle.
The formula to calculate the area of a sector is:
Area of sector = (θ/360°) x πr²,
where θ is the central angle of the sector and r is the radius of the circle.
We are given that the area of the sector is 210 square centimeters, and the central angle is 30°. So we can plug these values into the formula and solve for r.
210 = (30/360) x πr²
210 = (1/12)πr²
Now, we can solve for r.
Multiply both sides of the equation by 12:
12 * 210 = πr²
2520 = πr²
Divide both sides of the equation by π:
r² = 2520/π
Finally, take the square root of both sides to solve for r:
r ≈ √(2520/π)
Using a calculator, the approximate value of r is:
r ≈ 28.2 cm (rounded to the nearest tenth)
Now that we have the radius, we can calculate the area of the circle.
The formula for calculating the area of a circle is:
Area = πr²
Plugging in the value of the radius we found, we have:
Area = π(28.2)²
Calculating this expression, we get:
Area ≈ 2501.3 square cm (rounded to the nearest tenth)
Therefore, the area of the circle that contains the given sector is approximately 2501.3 square centimeters.
To find the area of the circle that contains the sector, we need to use the formula for the area of a sector. The formula for the area of a sector is:
Area of sector = (angle measure / 360°) * π * r^2
Given that the sector has an area of 210 square centimeters and an arc measure of 30°, we have:
210 = (30/360) * π * r^2
Simplifying the equation:
210 = (1/12) * π * r^2
Multiplying both sides by 12:
12 * 210 = π * r^2
2520 = π * r^2
Divide both sides by π:
2520 / π = r^2
Using the value of π as approximately 3.14159, we can calculate r^2:
2520 / 3.14159 = r^2
Using a calculator to find the value of 2520 divided by 3.14159, we get:
r^2 ≈ 802.847
To find the value of r, we take the square root of both sides:
r ≈ √802.847
Using a calculator to find the square root of 802.847, we get:
r ≈ 28.34
Now that we have the value of r, we can find the area of the circle using the formula:
Area of circle = π * r^2
Substituting the value of r, we get:
Area of circle ≈ 3.14159 * (28.34)^2
Using a calculator to find the value, we get:
Area of circle ≈ 2522.58
Rounding to the nearest tenth, the area of the circle that contains the sector is approximately 2522.6 square centimeters.