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To solve the problem, we need to find the area of the entire circle \( \odot T \), given that the area of the shaded sector (which corresponds to a central angle of 144°) is \( 4\pi \).

### Step-by-Step Solution

1. **Understand the relationship between the sector and the circle:**
   - The area of a sector of a circle is given by:
     \[
     \text{Area of Sector} = \frac{\theta}{360^\circ} \times \text{Area of the Circle}
     \]
     where \( \theta \) is the central angle in degrees.

2. **Set up the equation:**
   - For this problem, \( \theta = 144^\circ \), and the area of the shaded sector is \( 4\pi \). Let the area of the circle be \( A \).
     \[
     \frac{144^\circ}{360^\circ} \times A = 4\pi
     \]

3. **Solve for the area of the circle \( A \):**
   - Simplify the fraction:
     \[
     \frac{144^\circ}{360^\circ} = \frac{2}{5}
     \]
   - Now, substitute and solve for \( A \):
     \[
     \frac{2}{5} \times A = 4\pi
     \]
     \[
     A = 4\pi \times \frac{5}{2} = 10\pi
     \]

So, the area of \( \odot T \) is \( \boxed{10\pi} \).

Would you like more details on any of the steps? Here are some related questions you might find useful:

1. How is the area of a sector derived from the area of a circle?
2. What would the area of the circle be if the angle was different, say 90°?
3. How would the problem change if the shaded area was \(6\pi\) instead of \(4\pi\)?
4. What is the radius of the circle if the area is \(10\pi\)?
5. How does the central angle affect the area of the sector?

**Tip:** Remember that the area of a circle is proportional to the square of its radius, and the sector area is proportional to the central angle.

Use the diagram to answer the question: The area of the shaded sector is 4π. What is the area of ⊙T?

Discover how to calculate the total area of a circle when given the area of a shaded sector with a central angle of 144°. This step-by-step solution involves key geometric concepts and formulas, suitable for students in Grades 9-11.

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