https://r2.erweima.ai/mathbot/image/1725917213719ym4vbb9n.jpg

The image shows a sector of a circle with a radius \( R = 10 \, \text{cm} \) and a central angle of \( \frac{2\pi}{3} \) radians. We are asked to find the area of the yellow region.

### Step-by-step solution:

1. **Area of the full circle**:
   The area \( A_{\text{circle}} \) of a full circle is given by the formula:
   \[
   A_{\text{circle}} = \pi R^2
   \]
   Substituting \( \pi = 3.14 \) and \( R = 10 \, \text{cm} \):
   \[
   A_{\text{circle}} = 3.14 \times (10)^2 = 3.14 \times 100 = 314 \, \text{cm}^2
   \]

2. **Area of the sector**:
   The area of the sector is a fraction of the full circle, determined by the ratio of the central angle to the full angle of a circle (which is \( 2\pi \) radians). For a central angle of \( \frac{2\pi}{3} \), the fraction of the circle's area is:
   \[
   \frac{\theta}{2\pi} = \frac{\frac{2\pi}{3}}{2\pi} = \frac{1}{3}
   \]
   So, the area of the sector is:
   \[
   A_{\text{sector}} = \frac{1}{3} \times A_{\text{circle}} = \frac{1}{3} \times 314 = 104.67 \, \text{cm}^2
   \]

Thus, the area of the yellow region is \( \boxed{104.67 \, \text{cm}^2} \).

Would you like further details or clarifications?

### Related Questions:
1. How do you calculate the area of a sector with a different radius?
2. What is the formula for calculating the circumference of the circle?
3. How would the area change if the angle of the sector was \( \frac{\pi}{2} \)?
4. Can you explain how to derive the formula for the area of a sector?
5. What would the area of the sector be if the radius were doubled?

### Tip:
Always remember to adjust the formula for sectors by the ratio of the central angle to \( 2\pi \), as it gives the fraction of the circle.

Qual a área destacada em amarelo? Considere Pi = 3,14 e R = 10 cm.

This problem demonstrates how to calculate the area of a sector of a circle with a radius of 10 cm and a central angle of 2π/3 radians. It covers key geometric formulas like the area of a circle and the fraction of the sector.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation