If the diameter of a circle is increased by 200 % , then by what per cent has its circumference increased ?

Options

A. 1
B. 0.5
C. 2
D. 1.5

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the initial diameter
Let the initial diameter of the circle be \( d \).

### Step 2: Calculate the new diameter
If the diameter is increased by 200%, we need to calculate the new diameter. An increase of 200% means the new diameter is:
\[ \text{New Diameter} = d + 2d = 3d \]

### Step 3: Calculate the initial circumference
The formula for the circumference \( C \) of a circle is given by:
\[ C = \pi d \]
So, the initial circumference is:
\[ C_{\text{initial}} = \pi d \]

### Step 4: Calculate the new circumference
Now, we can find the new circumference using the new diameter:
\[ C_{\text{new}} = \pi \times \text{New Diameter} = \pi \times 3d = 3\pi d \]

### Step 5: Find the increase in circumference
To find the increase in circumference, we subtract the initial circumference from the new circumference:
\[ \text{Increase in Circumference} = C_{\text{new}} - C_{\text{initial}} = 3\pi d - \pi d = 2\pi d \]

### Step 6: Calculate the percentage increase
To find the percentage increase in circumference, we use the formula:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{\text{Original}} \times 100\%\right) \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left(\frac{2\pi d}{\pi d} \times 100\%\right) = 2 \times 100\% = 200\% \]

### Final Answer:
The circumference of the circle has increased by **200%**.

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