If the radius of a circle is increased three times, by how many times will the circumference increase ?

Answer

To solve the problem step by step, we will analyze how the circumference of a circle changes when the radius is increased three times.

### Step-by-Step Solution:

1. **Understand the relationship between radius and circumference**:
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

2. **Define the initial radius**:
Let the initial radius of the circle be \( R_1 \).

3. **Define the new radius after increase**:
If the radius is increased three times, the new radius \( R_2 \) can be expressed as:
\[ R_2 = 3R_1 \]

4. **Calculate the initial circumference**:
Using the initial radius \( R_1 \), the initial circumference \( C_1 \) is:
\[ C_1 = 2\pi R_1 \]

5. **Calculate the new circumference**:
Using the new radius \( R_2 \), the new circumference \( C_2 \) is:
\[ C_2 = 2\pi R_2 = 2\pi (3R_1) = 6\pi R_1 \]

6. **Determine the increase in circumference**:
To find out how many times the circumference has increased, we can set up the ratio of the new circumference to the initial circumference:
\[ \text{Increase factor} = \frac{C_2}{C_1} = \frac{6\pi R_1}{2\pi R_1} \]

7. **Simplify the ratio**:
Simplifying the fraction gives:
\[ \text{Increase factor} = \frac{6}{2} = 3 \]

8. **Conclusion**:
Therefore, the circumference of the circle increases by a factor of 3.

### Final Answer:
The circumference will increase by **3 times**.