https://file.aiquickdraw.com/mathbot/file/1727062554159zxwkzre7.jpg

We are given the area of a sector as \( 10\pi \, \text{square units} \), and the radius of the circle is 10 units. The task is to find the central angle \( \theta \) in degrees.

### Step-by-step Solution:

1. **Formula for the area of a sector**:
   \[
   A = \frac{\theta}{360^\circ} \cdot \pi r^2
   \]
   Where:
   - \( A \) is the area of the sector,
   - \( \theta \) is the central angle in degrees,
   - \( r \) is the radius of the circle.

2. **Substitute the known values**:
   - \( A = 10\pi \),
   - \( r = 10 \).

   Plugging these into the formula:
   \[
   10\pi = \frac{\theta}{360^\circ} \cdot \pi (10)^2
   \]
   Simplify:
   \[
   10\pi = \frac{\theta}{360^\circ} \cdot 100\pi
   \]
   Cancel \( \pi \) on both sides:
   \[
   10 = \frac{\theta}{360^\circ} \cdot 100
   \]
   Simplify further:
   \[
   10 = \frac{100\theta}{360^\circ}
   \]
   Multiply both sides by \( 360^\circ \):
   \[
   10 \times 360^\circ = 100\theta
   \]
   \[
   3600^\circ = 100\theta
   \]
   Solve for \( \theta \):
   \[
   \theta = \frac{3600^\circ}{100} = 36^\circ
   \]

Thus, the central angle \( \theta \) is \( \boxed{36^\circ} \).

Would you like further details on any step?

### Related Questions:
1. What is the formula for the area of a sector in radians?
2. How does the area of a sector change if the central angle doubles?
3. What is the total area of the circle given in this problem?
4. How can we calculate the length of the arc for this sector?
5. How would the central angle change if the area of the sector was \(20\pi\)?

### Tip:
Always simplify by canceling common factors (like \( \pi \)) early in sector-related formulas to make calculations easier.

Find the measure (in degrees) of the central angle. The area of a sector is 10π sq units. The radius of the circle is 10 units.

Learn how to find the measure of the central angle when given the sector's area (10π square units) and the circle's radius (10 units). This problem utilizes the formula for the area of a sector, and is ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation