Find the circumference of a circle if the area of a quadrant of the circle is 154cm^2

Answer

To find the circumference of a circle when the area of a quadrant of the circle is given as 154 cm², we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Area of a Quadrant**:
The area of a quadrant of a circle is given by the formula:
\[ \text{Area of Quadrant} = \frac{\pi r^2}{4} \]
where \( r \) is the radius of the circle.

2. **Set Up the Equation**:
We know the area of the quadrant is 154 cm², so we can set up the equation:
\[ \frac{\pi r^2}{4} = 154 \]

3. **Multiply Both Sides by 4**:
To eliminate the fraction, multiply both sides of the equation by 4:
\[ \pi r^2 = 154 \times 4 \]
\[ \pi r^2 = 616 \]

4. **Substitute the Value of π**:
Use the approximate value of π as \( \frac{22}{7} \):
\[ \frac{22}{7} r^2 = 616 \]

5. **Multiply Both Sides by 7**:
To get rid of the fraction, multiply both sides by 7:
\[ 22 r^2 = 616 \times 7 \]
\[ 22 r^2 = 4312 \]

6. **Divide by 22**:
Now, divide both sides by 22 to solve for \( r^2 \):
\[ r^2 = \frac{4312}{22} \]
\[ r^2 = 196 \]

7. **Take the Square Root**:
Now, take the square root of both sides to find \( r \):
\[ r = \sqrt{196} = 14 \text{ cm} \]

8. **Calculate the Circumference**:
The circumference \( C \) of the circle is given by the formula:
\[ C = 2\pi r \]
Substituting the value of \( r \) and \( \pi \):
\[ C = 2 \times \frac{22}{7} \times 14 \]

9. **Simplify the Expression**:
\[ C = 2 \times 22 \times 2 = 88 \text{ cm} \]

### Final Answer:
The circumference of the circle is **88 cm**.

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