The sector *COB* is cut from the circle with center *O*. The ratio of the area of the sector removed from the whole circle to the area of the sector removed from the inner circle is ___

To solve this problem, we need to find the ratio of the area of the sector removed from the entire circle to the area of the sector removed from the inner circle.

Let's define the following:

- \( r_1 \) as the radius of the smaller circle (inner circle).
- \( r_2 \) as the radius of the larger circle (whole circle).
- \( \theta \) as the angle subtended by the sector \( COB \) at the center \( O \) (in radians).

### 1. **Area of the Sector from the Entire Circle:**
The area of a sector of a circle is given by:

\[
\text{Area of sector in larger circle} = \frac{1}{2} r_2^2 \theta
\]

### 2. **Area of the Sector from the Inner Circle:**
Similarly, the area of the sector from the inner circle is:

\[
\text{Area of sector in inner circle} = \frac{1}{2} r_1^2 \theta
\]

### 3. **Ratio of the Areas:**
To find the ratio of these two areas:

\[
\text{Ratio} = \frac{\text{Area of sector in larger circle}}{\text{Area of sector in inner circle}} = \frac{\frac{1}{2} r_2^2 \theta}{\frac{1}{2} r_1^2 \theta}
\]

The \( \frac{1}{2} \) and \( \theta \) terms cancel out, leaving us with:

\[
\text{Ratio} = \frac{r_2^2}{r_1^2}
\]

### Conclusion:
The ratio of the area of the sector removed from the whole circle to the area of the sector removed from the inner circle is \( \frac{r_2^2}{r_1^2} \).

Would you like any additional details on this problem or its concepts?

### Related Questions:
1. What is the formula for the area of a circle?
2. How does the central angle \( \theta \) affect the area of a sector?
3. Can this ratio be simplified if \( r_2 = 2r_1 \)?
4. What is the area of the annular region between the two circles?
5. How would the answer change if the circles were not concentric?
6. How is the formula for the sector area derived from the area of a full circle?
7. What would be the ratio if only the difference in areas of sectors was considered?
8. How can you express this ratio in terms of percentages?

### Tip:
Always simplify ratios by canceling common terms whenever possible to make your final answer more intuitive and easier to interpret.

Explore the ratio of the area of a sector removed from a whole circle to that removed from an inner circle in concentric circles. Learn how to calculate this ratio using the area formula for circle sectors. Suitable for high school students studying geometry.

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