A circle is inscribed in a quadrant of a larger circle

whose radius is 4 cm. Find the area of the inscribed

circle.

A. 9.86 cm2 C. 8.62 cm2

B. 7.33 cm2 D. 12.57 cm2

To find the area of the inscribed circle in the given scenario, let's break down the problem:

- We have a quadrant of a larger circle, which is one-fourth of a circle.
- The radius of the larger circle is given as 4 cm.

### Step-by-step solution:

1. **Position and Geometry:**
   - The inscribed circle touches both the radii and the arc of the quadrant.
   - The radius of the larger circle is \( R = 4 \) cm.

2. **Relationship between the Radii:**
   The radius \( r \) of the smaller, inscribed circle in the quadrant can be derived geometrically. In this case, the radius \( r \) of the inscribed circle can be found using the formula for a circle inscribed in a quadrant of a circle:
   \[
   r = \frac{R}{2}
   \]
   where \( R \) is the radius of the larger circle. Thus,
   \[
   r = \frac{4}{2} = 2 \text{ cm}
   \]

3. **Area of the Inscribed Circle:**
   The area \( A \) of a circle is given by the formula:
   \[
   A = \pi r^2
   \]
   Substituting \( r = 2 \text{ cm} \):
   \[
   A = \pi (2)^2 = 4\pi
   \]

   Using \( \pi \approx 3.1416 \):
   \[
   A = 4 \times 3.1416 = 12.566 \, \text{cm}^2
   \]

Thus, the area of the inscribed circle is approximately **12.57 cm²**.

### The correct answer is D. \( 12.57 \, \text{cm}^2 \).

Would you like further details on this? Here are five related questions:

1. How is the formula for a circle inscribed in a quadrant derived?
2. What is the area of the larger circle from which the quadrant is taken?
3. How would the solution change if the radius of the larger circle was different?
4. Can the formula for the inscribed circle's radius apply to other shapes, such as a semicircle?
5. How do we calculate the perimeter of the quadrant?

**Tip:** Always remember that the area of a circle is proportional to the square of its radius, which makes small changes in radius result in significant changes in area.

A circle is inscribed in a quadrant of a larger circle whose radius is 4 cm. Find the area of the inscribed circle. A. 9.86 cm² C. 8.62 cm² B. 7.33 cm² D. 12.57 cm²

Learn how to calculate the area of a circle inscribed in a quadrant of a larger circle with radius 4 cm. This geometry problem uses key concepts such as the relationship between the radii of circles and the area formula, suitable for Grades 8-10.

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