Problem Overview:
The task involves finding the area of the shaded region in each of the given concentric circles. The formula for the area of a circle is:
\[
\text{Area of a circle} = \pi r^2
\]
For concentric circles, the area of the shaded region is the difference between the area of the outer circle and the area of the inner circle:
\[
\text{Area of shaded region} = \pi R^2 - \pi r^2 = \pi (R^2 - r^2)
\]
Where:
- \( R \) is the radius of the outer circle.
- \( r \) is the radius of the inner circle.
We are given that \(\pi = \frac{22}{7}\) or 3.14. We will use this value to calculate the areas.
---
Step-by-Step Solution:
####
Problem 1:
- Outer radius (\( R \)) = 8 mm
- Inner radius (\( r \)) = 6 mm
Using the formula for the shaded area:
\[
\text{Area of shaded region} = \pi (R^2 - r^2)
\]
Substitute the values:
\[
\text{Area} = \pi (8^2 - 6^2) = \pi (64 - 36) = \pi (28)
\]
Using \(\pi = \frac{22}{7}\):
\[
\text{Area} = \frac{22}{7} \times 28 = 22 \times 4 = 88 \, \text{mm}^2
\]
Answer for Problem 1:
\[
\boxed{88}
\]
---
####
Problem 2:
- Outer radius (\( R \)) = 9 cm
- Inner radius (\( r \)) = 5 cm
Using the formula for the shaded area:
\[
\text{Area of shaded region} = \pi (R^2 - r^2)
\]
Substitute the values:
\[
\text{Area} = \pi (9^2 - 5^2) = \pi (81 - 25) = \pi (56)
\]
Using \(\pi = \frac{22}{7}\):
\[
\text{Area} = \frac{22}{7} \times 56 = 22 \times 8 = 176 \, \text{cm}^2
\]
Answer for Problem 2:
\[
\boxed{176}
\]
---
####
Problem 3:
- Outer radius (\( R \)) = 8 cm
- Inner radius (\( r \)) = 5 cm
Using the formula for the shaded area:
\[
\text{Area of shaded region} = \pi (R^2 - r^2)
\]
Substitute the values:
\[
\text{Area} = \pi (8^2 - 5^2) = \pi (64 - 25) = \pi (39)
\]
Using \(\pi = \frac{22}{7}\):
\[
\text{Area} = \frac{22}{7} \times 39 = \frac{858}{7} \approx 122.57 \, \text{cm}^2
\]
Answer for Problem 3:
\[
\boxed{122.57}
\]
---
####
Problem 4:
- Outer radius (\( R \)) = 4.5 m
- Inner radius (\( r \)) = 3 m
Using the formula for the shaded area:
\[
\text{Area of shaded region} = \pi (R^2 - r^2)
\]
Substitute the values:
\[
\text{Area} = \pi (4.5^2 - 3^2) = \pi (20.25 - 9) = \pi (11.25)
\]
Using \(\pi = 3.14\):
\[
\text{Area} = 3.14 \times 11.25 = 35.325 \, \text{m}^2
\]
Answer for Problem 4:
\[
\boxed{35.325}
\]
---
Final Answers:
1. \(\boxed{88}\)
2. \(\boxed{176}\)
3. \(\boxed{122.57}\)
4. \(\boxed{35.325}\)
Parent Tip: Review the logic above to help your child master the concept of area of circles worksheet pdf.
