This exercise involves the formula for the area of a circular sector.

The area of a sector of a circle with a central angle of 120° is

71 m2.

Find the radius of the circle. (Round your answer to one decimal place.)
m

To find the radius of the circle given the area of a sector, we will use the formula for the area of a sector of a circle. The formula for the area \(A\) of a sector with a central angle \(\theta\) (in degrees) and radius \(r\) is:

\[
A = \frac{\theta}{360} \times \pi r^2
\]

In this problem:
- \(A = 71 \, \text{m}^2\) (area of the sector)
- \(\theta = 120^\circ\) (central angle)

We want to find the radius \(r\) of the circle.

### Step-by-Step Solution

1. Substitute the given values into the formula:

\[
71 = \frac{120}{360} \times \pi r^2
\]

2. Simplify the fraction:

\[
71 = \frac{1}{3} \times \pi r^2
\]

3. Multiply both sides by 3 to eliminate the fraction:

\[
213 = \pi r^2
\]

4. Solve for \(r^2\) by dividing both sides by \(\pi\):

\[
r^2 = \frac{213}{\pi}
\]

5. Calculate the value:

\[
r^2 \approx \frac{213}{3.1416} \approx 67.8
\]

6. Take the square root of both sides to solve for \(r\):

\[
r \approx \sqrt{67.8} \approx 8.2
\]

### Final Answer

The radius of the circle is approximately \(8.2\) meters.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the formula change if the central angle was given in radians instead of degrees?
2. What is the area of a full circle with the same radius?
3. How would the area of the sector change if the radius was doubled while keeping the central angle the same?
4. What is the length of the arc of this sector?
5. How does the area of a sector relate to the area of the corresponding segment of a circle?

**Tip:** Always remember to convert the central angle to radians if you are using a formula that requires radians!

Learn how to find the radius of a circle sector with a 120° central angle using the area formula. Detailed step-by-step solution provided. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation