Circle Formulas

lessonbunny.com

Answer Key & Explanations

Circle Formulas · Grade 7 · Worksheet 1

1. A circular swimming pool has a diameter of 14 meters. A concrete walkway extends 2 meters outward from the edge of the pool all the way around. What is the total area of the walkway? (Use π = 3.14) Answer: 100.48 Solution: - Pool diameter = 14 m → pool radius = 14 / 2 = 7 m - Walkway extends 2 m outward from the edge of the pool all around. - So the outer edge of the walkway forms a larger circle.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have: - Pool diameter = 14 m → pool radius = 14 / 2 = 7 m - Walkway extends 2 m outward from the edge of the pool all around. - So the outer edge of the walkway forms a larger circle. --- **Step 2: Find the radii** - Inner radius (pool only): \( r_1 = 7 \) m - Outer radius (pool + walkway): \( r_2 = 7 + 2 = 9 \) m --- **Step 3: Area of the larger circle (pool + walkway)** Area of a circle = π × radius² Area of larger circle = π × (9)² = π × 81 --- **Step 4: Area of the pool (inner circle)** Area of inner circle = π × (7)² = π × 49 --- **Step 5: Area of the walkway** Walkway area = Area of larger circle − Area of inner circle = π × 81 − π × 49 = π × (81 − 49) = π × 32 --- **Step 6: Substitute π = 3.14** Walkway area = 3.14 × 32 3.14 × 30 = 94.2 3.14 × 2 = 6.28 Sum = 94.2 + 6.28 = 100.48 --- **Step 7: Final answer** Total area of the walkway = 100.48 square meters.

2. Liam is designing a circular fountain for his community garden. The fountain will have a diameter of 14 feet. He needs to purchase a decorative border for the outer edge and special tiles to cover the entire surface area. How many feet of border does he need for the circumference, and what is the area in square feet that needs to be tiled? Answer: Circumference: 43.96 feet, Area: 153.86 square feet Solution: The fountain is circular. Diameter = 14 feet. Radius = Diameter / 2 Radius = 14 / 2 = 7 feet.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Identify the given information** The fountain is circular. Diameter = 14 feet. --- **Step 2: Find the radius** Radius = Diameter / 2 Radius = 14 / 2 = 7 feet. --- **Step 3: Calculate the circumference (border length)** Circumference formula: C = 2 * π * r We use π ≈ 3.14. C = 2 * 3.14 * 7 C = 6.28 * 7 C = 43.96 feet. So, the border needed is **43.96 feet**. --- **Step 4: Calculate the area (tiling surface)** Area formula: A = π * r^2 A = 3.14 * (7)^2 A = 3.14 * 49 A = 153.86 square feet. So, the tiling area is **153.86 square feet**. --- **Final Answer:** Circumference: 43.96 feet Area: 153.86 square feet

3. A circular garden has a radius of 9 meters. What is its circumference? Use π = 3.14. Answer: 56.52 meters Solution: Write down the formula for circumference: C = 2πr. Substitute the given values: r = 9 meters, π = 3.14. Calculate: C = 2 × 3.14 × 9.
   Full step-by-step solution

   Step 1: Write down the formula for circumference: C = 2πr. Step 2: Substitute the given values: r = 9 meters, π = 3.14. Step 3: Calculate: C = 2 × 3.14 × 9. Step 4: First multiply 2 × 9 = 18. Step 5: Then multiply 18 × 3.14 = 56.52. Step 6: The circumference is 56.52 meters.

4. π × (18² - 12²) = ? Answer: 180π Solution: Calculate the square of the larger radius: 18² = 324 Calculate the square of the smaller radius: 12² = 144 Subtract the smaller square from the larger square: 324 - 144 = 180 Multiply the result by π: 180 × π = 180π The final answer is 180π.
   Full step-by-step solution

   Step 1: Calculate the square of the larger radius: 18² = 324 Step 2: Calculate the square of the smaller radius: 12² = 144 Step 3: Subtract the smaller square from the larger square: 324 - 144 = 180 Step 4: Multiply the result by π: 180 × π = 180π Step 5: The final answer is 180π.