https://file.aiquickdraw.com/mathbot/file/1727003880296uny63x1e.jpg

The problem asks to find the area of the shaded region, given two congruent circles drawn inside a larger circle. Here’s how we can solve it:

### Information provided:
1. The area of the large circle is \( 324\pi \) square centimeters.
2. The figure shows that two smaller congruent circles are inscribed within the larger circle.
3. The radius of each smaller circle is denoted as \( r \).

### Step-by-step solution:

1. **Find the radius of the large circle**:  
   The area of the large circle is given by the formula:
   \[
   \text{Area of large circle} = \pi R^2 = 324\pi
   \]
   Solving for \( R \):
   \[
   R^2 = 324 \quad \Rightarrow \quad R = \sqrt{324} = 18 \, \text{cm}
   \]

2. **Relationship between large and small circles**:
   The two smaller congruent circles are placed inside the larger circle and touch each other at one point. Together, their diameters form the diameter of the larger circle. So:
   \[
   2r = 18 \quad \Rightarrow \quad r = \frac{18}{2} = 9 \, \text{cm}
   \]

3. **Find the area of each smaller circle**:  
   The area of a smaller circle is:
   \[
   \text{Area of small circle} = \pi r^2 = \pi (9^2) = 81\pi \, \text{cm}^2
   \]
   Since there are two smaller circles, the total area of the two smaller circles is:
   \[
   \text{Total area of two smaller circles} = 2 \times 81\pi = 162\pi \, \text{cm}^2
   \]

4. **Find the area of the shaded region**:  
   The shaded area is the area of the large circle minus the combined area of the two smaller circles:
   \[
   \text{Shaded area} = \text{Area of large circle} - \text{Total area of two smaller circles}
   \]
   Substituting the known values:
   \[
   \text{Shaded area} = 324\pi - 162\pi = 162\pi \, \text{cm}^2
   \]

Thus, the area of the shaded region is \( 162\pi \, \text{cm}^2 \), which numerically is approximately:
\[
162\pi \approx 162 \times 3.1416 = 508.68 \, \text{cm}^2
\]

If you have any questions or need further details, feel free to ask! Here are some related questions:

1. How would the problem change if the smaller circles were not congruent?
2. Can we calculate the perimeter of the shaded region?
3. What would happen if there were more than two smaller circles inscribed?
4. How does the relationship between the radius of the large circle and the smaller circles help simplify the problem?
5. Can this technique be applied to other shapes, like squares or triangles?

**Tip**: Always start by calculating the radii when dealing with circle area problems!

Two congruent circles are drawn inside a big circle as shown. If the area of the big circle is 324π square centimeters, find the area of the shaded portion.

Solve a geometry problem involving two congruent circles inside a larger circle. Learn how to calculate the shaded area given the area of the large circle as 324π cm². This step-by-step guide uses circle area formulas and geometric properties, suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation