Step 1: Find the diameter: Diameter = 2 × radius Diameter = 2 × 7 Diameter = 14 cm

Step 2: Find the circumference: Circumference = 2πr (or πd) C = 2π(7) C = 14π cm

Step 3: Approximate value (optional): C ≈ 14 × 3.14159 C ≈ 43.98 cm

Answer: Diameter = 14 cm, Circumference = 14π cm (≈ 43.98 cm)

Circumference: $C = 2\pi r = 2\pi(5) = 10\pi$

Area: $A = \pi r^2 = \pi(5)^2 = 25\pi$

Answer: Circumference = $10\pi$ (or ≈ 31.4), Area = $25\pi$ (or ≈ 78.5)

Step 1: Use the circumference formula: C = 2πr

Step 2: Substitute known values: 31.4 = 2πr 31.4 = 2(3.14)r 31.4 = 6.28r

Step 3: Solve for r: r = 31.4 / 6.28 r = 5 cm

Step 4: Verify: C = 2π(5) = 10π ≈ 10(3.14) = 31.4 ✓

Answer: The radius is 5 cm

Step 1: Find the radius $r = \frac{d}{2} = \frac{16}{2} = 8$

Step 2: Use arc length formula $\text{Arc length} = \frac{\theta}{360°} \times 2\pi r$

$= \frac{45}{360} \times 2\pi(8)$

$= \frac{1}{8} \times 16\pi$

$= 2\pi$

Answer: Arc length is $2\pi$ (or ≈ 6.28)

Step 1: Use the area formula: Area = πr²

Step 2: Substitute the known area: 49π = πr²

Step 3: Solve for r²: 49 = r² r = √49 r = 7

Step 4: Find the circumference: C = 2πr C = 2π(7) C = 14π

Step 5: Verify the area: A = π(7)² = 49π ✓

Answer: Radius = 7, Circumference = 14π

Step 1: Understand the problem: Need to find the area between two circles Area of ring = Area of large circle - Area of small circle

Step 2: Find area of large circle: A_large = πr² A_large = π(8)² A_large = 64π cm²

Step 3: Find area of small circle: A_small = πr² A_small = π(5)² A_small = 25π cm²

Step 4: Find area of ring: Area of ring = 64π - 25π Area of ring = 39π cm²

Step 5: Approximate (optional): 39π ≈ 39 × 3.14159 ≈ 122.52 cm²

Answer: The area of the ring is 39π cm² (≈ 122.52 cm²)

Draw a radius to the chord's endpoint and a perpendicular from center to chord.

This creates a right triangle:

• Hypotenuse = radius = 10
• One leg = distance from center = 8
• Other leg = half the chord length

Use Pythagorean Theorem: $8^2 + \left(\frac{c}{2}\right)^2 = 10^2$

$64 + \frac{c^2}{4} = 100$

$\frac{c^2}{4} = 36$

$c^2 = 144$

$c = 12$

Answer: The chord length is 12 cm

Step 1: Find the radius of the garden: Diameter = 20 m Radius of garden = 20/2 = 10 m

Step 2: Find area of the garden: A_garden = πr² A_garden = π(10)² A_garden = 100π m²

Step 3: Find the outer radius (garden + path): Path width = 2 m Outer radius = 10 + 2 = 12 m

Step 4: Find total area (garden + path): A_total = π(12)² A_total = 144π m²

Step 5: Find area of just the path: A_path = A_total - A_garden A_path = 144π - 100π A_path = 44π m²

Step 6: Approximate values: Garden: 100π ≈ 314.16 m² Path: 44π ≈ 138.23 m² Total: 144π ≈ 452.39 m²

Step 7: Verify: Garden + Path = 100π + 44π = 144π ✓

Answer: (a) Garden area = 100π m² ≈ 314.16 m² (b) Path area = 44π m² ≈ 138.23 m² (c) Total area = 144π m² ≈ 452.39 m²