Problem Overview:
The task involves calculating the areas of circles and sectors. We are given specific dimensions (radii or diameters) and angles for each shape. The formula for the area of a circle is:
\[
\text{Area of a Circle} = \pi r^2
\]
For sectors, the area is calculated as a fraction of the circle's area based on the central angle:
\[
\text{Area of a Sector} = \frac{\theta}{360^\circ} \times \pi r^2
\]
where \( \theta \) is the central angle in degrees.
We will use \( \pi = 3.14 \) as specified.
---
Step-by-Step Solution:
####
Part 1: Area of Circles
1.
Circle 1:
- Radius \( r = 18 \) yd
- Area:
\[
\text{Area} = \pi r^2 = 3.14 \times (18)^2 = 3.14 \times 324 = 1017.36 \text{ yd}^2
\]
2.
Circle 2:
- Diameter \( d = 28 \) ft
- Radius \( r = \frac{d}{2} = \frac{28}{2} = 14 \) ft
- Area:
\[
\text{Area} = \pi r^2 = 3.14 \times (14)^2 = 3.14 \times 196 = 615.44 \text{ ft}^2
\]
3.
Circle 3:
- Radius \( r = 5.6 \) m
- Area:
\[
\text{Area} = \pi r^2 = 3.14 \times (5.6)^2 = 3.14 \times 31.36 = 98.46 \text{ m}^2
\]
---
####
Part 2: Area of Shaded Sectors
4.
Sector 4:
- Radius \( r = 3.5 \) yd
- Central angle \( \theta = 40^\circ \)
- Area of the sector:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 = \frac{40}{360} \times 3.14 \times (3.5)^2
\]
\[
= \frac{1}{9} \times 3.14 \times 12.25 = \frac{1}{9} \times 38.465 = 4.274 \text{ yd}^2
\]
5.
Sector 5:
- Radius \( r = 7.5 \) ft
- Central angle \( \theta = 70^\circ \)
- Area of the sector:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 = \frac{70}{360} \times 3.14 \times (7.5)^2
\]
\[
= \frac{7}{36} \times 3.14 \times 56.25 = \frac{7}{36} \times 176.625 = 34.729 \text{ ft}^2
\]
6.
Sector 6:
- Radius \( r = 12 \) m
- Central angle \( \theta = 90^\circ \)
- Area of the sector:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 = \frac{90}{360} \times 3.14 \times (12)^2
\]
\[
= \frac{1}{4} \times 3.14 \times 144 = \frac{1}{4} \times 452.16 = 113.04 \text{ m}^2
\]
7.
Sector 7:
- Radius \( r = 6 \) ft
- Central angle \( \theta = 120^\circ \)
- Area of the sector:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 = \frac{120}{360} \times 3.14 \times (6)^2
\]
\[
= \frac{1}{3} \times 3.14 \times 36 = \frac{1}{3} \times 113.04 = 37.68 \text{ ft}^2
\]
8.
Sector 8:
- Radius \( r = 3 \) m
- Central angle \( \theta = 205^\circ \)
- Area of the sector:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 = \frac{205}{360} \times 3.14 \times (3)^2
\]
\[
= \frac{205}{360} \times 3.14 \times 9 = \frac{205}{360} \times 28.26 = 16.01 \text{ m}^2
\]
---
Final Answers:
1. Circle 1: \( \boxed{1017.36} \) yd²
2. Circle 2: \( \boxed{615.44} \) ft²
3. Circle 3: \( \boxed{98.46} \) m²
4. Sector 4: \( \boxed{4.27} \) yd²
5. Sector 5: \( \boxed{34.73} \) ft²
6. Sector 6: \( \boxed{113.04} \) m²
7. Sector 7: \( \boxed{37.68} \) ft²
8. Sector 8: \( \boxed{16.01} \) m²
---
Boxed Final Answer:
\[
\boxed{1017.36, 615.44, 98.46, 4.27, 34.73, 113.04, 37.68, 16.01}
\]
Parent Tip: Review the logic above to help your child master the concept of areas of circles and sectors worksheet.
