Circle Formulas

lessonbunny.com

Answer Key & Explanations

Circle Formulas · Grade 7 · Worksheet 2

1. A circular fountain in a park has a diameter of 14 meters. A concrete walkway surrounds the fountain, extending 3 meters outward from the fountain's edge. What is the total area of the walkway? (Use π = 3.14) Answer: 160.14 Solution: The fountain is a circle with diameter = 14 m. The walkway extends 3 m outward from the edge of the fountain. Area of a circle = π × r² Area of fountain = π × (7)² = 3.14 × 49 = 153.86 m².
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** The fountain is a circle with diameter = 14 m. So the radius of the fountain = diameter / 2 = 14 / 2 = 7 m. The walkway extends 3 m outward from the edge of the fountain. So the outer radius (fountain + walkway) = 7 + 3 = 10 m. --- **Step 2: Area of the fountain** Area of a circle = π × r² Area of fountain = π × (7)² = 3.14 × 49 = 153.86 m². --- **Step 3: Total area (fountain + walkway)** Total radius = 10 m. Total area = π × (10)² = 3.14 × 100 = 314 m². --- **Step 4: Area of the walkway only** Walkway area = Total area − Fountain area = 314 − 153.86 = 160.14 m². --- **Final Answer:** 160.14

2. A city is building a circular skate park with a diameter of 28 meters. The planners need to install a protective railing around the entire outer edge and calculate how much special rubber surfacing material is needed to cover the entire circular area. What is the total length of railing required (circumference) and how many square meters of surfacing material are needed (area)? Use π ≈ 3.14 and round your answers to the nearest whole number. Answer: 88 meters, 616 square meters Solution: Find the radius of the skate park. The diameter is 28 meters, so the radius is half of that: 28 ÷ 2 = 14 meters. Calculate the circumference (length of railing needed).
   Full step-by-step solution

   Step 1: Find the radius of the skate park. The diameter is 28 meters, so the radius is half of that: 28 ÷ 2 = 14 meters. Step 2: Calculate the circumference (length of railing needed). The formula is C = π × diameter. Using π ≈ 3.14: C = 3.14 × 28 = 87.92 meters. Rounded to the nearest whole number: 88 meters. Step 3: Calculate the area (amount of surfacing material needed). The formula is A = π × radius². Using π ≈ 3.14 and radius = 14 meters: A = 3.14 × (14 × 14) = 3.14 × 196 = 615.44 square meters. Rounded to the nearest whole number: 616 square meters. Step 4: Combine the answers. The city needs 88 meters of railing and 616 square meters of surfacing material.

3. π × (16² - 11²) = ? Answer: 135π Solution: Calculate 16² = 256 Calculate 11² = 121 Find the difference: 256 - 121 = 135 Multiply by π: 135 × π = 135π The answer is 135π.
   Full step-by-step solution

   Step 1: Calculate 16² = 256 Step 2: Calculate 11² = 121 Step 3: Find the difference: 256 - 121 = 135 Step 4: Multiply by π: 135 × π = 135π The answer is 135π.

4. A circular fountain has a radius of 13 meters. Find its circumference and area. Use π = 3.14. Answer: Circumference = 81.64 meters, Area = 530.66 square meters Solution: Write the formulas. Circumference C = 2πr, Area A = πr². Substitute r = 13 and π = 3.14.
   Full step-by-step solution

   Step 1: Write the formulas. Circumference C = 2πr, Area A = πr². Step 2: Substitute r = 13 and π = 3.14. Step 3: Calculate circumference: C = 2 × 3.14 × 13 = 2 × 3.14 = 6.28, then 6.28 × 13 = 81.64 meters. Step 4: Calculate area: A = 3.14 × 13² = 3.14 × 169 = 530.66 square meters. The answer is circumference = 81.64 meters, area = 530.66 square meters.