a sector with an area of 26 pi squared centimeter has a radius of 6 cm what is the perimeter of the sector of a circle in centimeter

To find the perimeter of the sector, we need to determine the arc length of the sector first, and then add the lengths of the two radii.

1. **Area of the sector**:
   \[
   \text{Area of the sector} = \frac{1}{2} r^2 \theta
   \]
   where \( r \) is the radius and \( \theta \) is the central angle in radians.

   Given:
   \[
   \frac{1}{2} \times 6^2 \times \theta = 26\pi
   \]
   Simplify and solve for \( \theta \):
   \[
   18 \theta = 26\pi
   \]
   \[
   \theta = \frac{26\pi}{18} = \frac{13\pi}{9} \, \text{radians}
   \]

2. **Arc length of the sector**:
   \[
   \text{Arc length} = r \theta = 6 \times \frac{13\pi}{9} = \frac{78\pi}{9} = \frac{26\pi}{3} \, \text{cm}
   \]

3. **Perimeter of the sector**:
   The perimeter of the sector includes the arc length and the two radii:
   \[
   \text{Perimeter} = \text{Arc length} + 2 \times \text{Radius}
   \]
   \[
   \text{Perimeter} = \frac{26\pi}{3} + 2 \times 6 = \frac{26\pi}{3} + 12 \, \text{cm}
   \]

So, the perimeter of the sector is:
\[
\boxed{\frac{26\pi}{3} + 12 \, \text{cm}}
\]

Would you like more details or have any questions?

Here are some more questions you might find interesting:
1. How do you find the area of a sector of a circle given the radius and the central angle?
2. How do you convert between degrees and radians?
3. What is the formula for the circumference of a circle?
4. How do you find the length of an arc of a circle?
5. How do you solve for the radius of a circle if the area is known?

**Tip**: When solving geometry problems involving circles, always keep track of whether angles are given in degrees or radians and convert if necessary.

Find the perimeter of a circle sector with a radius of 6 cm and an area of 26π cm². Learn how to calculate the arc length and perimeter using geometric formulas. Suitable for high school students studying geometry.

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