The ratio between the area of two circles is 4:7. What will be the ratio of their radii ?

Options

A. `2:sqrt(7)`
B. `4:7`
C. `16:49`
D. `4:sqrt(7)`

Answer

To find the ratio of the radii of two circles given the ratio of their areas, we can follow these steps:

### Step 1: Understand the relationship between area and radius
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

### Step 2: Set up the ratio of the areas
We are given that the ratio of the areas of two circles is:
\[ \frac{A_1}{A_2} = \frac{4}{7} \]
Let \( A_1 \) be the area of the first circle and \( A_2 \) be the area of the second circle.

### Step 3: Express the areas in terms of their radii
Using the formula for the area of a circle, we can express the areas in terms of their radii:
\[ A_1 = \pi r_1^2 \quad \text{and} \quad A_2 = \pi r_2^2 \]
where \( r_1 \) and \( r_2 \) are the radii of the first and second circles, respectively.

### Step 4: Substitute the areas into the ratio
Substituting the expressions for the areas into the ratio gives:
\[ \frac{\pi r_1^2}{\pi r_2^2} = \frac{4}{7} \]
The \( \pi \) cancels out from both sides:
\[ \frac{r_1^2}{r_2^2} = \frac{4}{7} \]

### Step 5: Take the square root of both sides
To find the ratio of the radii, we take the square root of both sides:
\[ \frac{r_1}{r_2} = \sqrt{\frac{4}{7}} = \frac{\sqrt{4}}{\sqrt{7}} = \frac{2}{\sqrt{7}} \]

### Step 6: Rationalize the denominator (optional)
To express the ratio in a more standard form, we can rationalize the denominator:
\[ \frac{2}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{2\sqrt{7}}{7} \]

### Final Answer
Thus, the ratio of the radii of the two circles is:
\[ \frac{r_1}{r_2} = \frac{2\sqrt{7}}{7} \]

---