Area and Surface Formulas

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Answer Key & Explanations

Area and Surface Formulas · Grade 7 · Worksheet 3

1. Emma is designing a cylindrical compost bin for her community garden. The bin has a radius of 0.7 meters and a height of 1.5 meters. She needs to paint the entire outer surface, including the top and the bottom, with weatherproof paint. What is the total surface area of the cylindrical compost bin in square meters? Use π ≈ 3.14 and round your answer to the nearest hundredth. Answer: 9.67 Solution: Identify the formula for the surface area of a cylinder: SA = 2πr² + 2πrh, where r is the radius and h is the height.
   Full step-by-step solution

   Step 1: Identify the formula for the surface area of a cylinder: SA = 2πr² + 2πrh, where r is the radius and h is the height. Step 2: Calculate the area of the two circular ends: 2 × π × r² = 2 × 3.14 × (0.7)² = 2 × 3.14 × 0.49 = 2 × 1.5386 = 3.0772 m². Step 3: Calculate the area of the curved side: 2πrh = 2 × 3.14 × 0.7 × 1.5 = 2 × 3.14 × 1.05 = 2 × 3.297 = 6.594 m². Step 4: Add the areas together: 3.0772 + 6.594 = 9.6712 m². Step 5: Round to the nearest hundredth: 9.67 m². The total surface area to be painted is 9.67 square meters.

2. A large cylindrical water storage tank at a factory has a diameter of 32 meters and a height of 18 meters. The tank needs to be painted on its entire exterior surface, including the top and bottom. Calculate the total surface area of the tank in square meters. Use π = 3.14 in your calculations. Answer: 3416.32 Solution: Find the radius of the cylinder. The diameter is 32 meters, so the radius is half of that: r = 32 ÷ 2 = 16 meters. Calculate the area of one circular end: πr² = 3.14 × (16)² = 3.14 × 256 = 803.84 square meters.
   Full step-by-step solution

   Step 1: Find the radius of the cylinder. The diameter is 32 meters, so the radius is half of that: r = 32 ÷ 2 = 16 meters. Step 2: Calculate the area of one circular end: πr² = 3.14 × (16)² = 3.14 × 256 = 803.84 square meters. Step 3: Since there are two circular ends (top and bottom), multiply by 2: 803.84 × 2 = 1607.68 square meters. Step 4: Calculate the curved surface area: 2πrh = 2 × 3.14 × 16 × 18 = 2 × 3.14 × 288 = 2 × 904.32 = 1808.64 square meters. Step 5: Add all the surface areas together: 1607.68 + 1808.64 = 3416.32 square meters. The total surface area is 3416.32 square meters.

3. Sophia is painting a cylindrical storage tank for a community art project. The tank has a radius of 7 feet and a height of 12 feet. She needs to paint the entire outer surface, including the top and the bottom. Each can of paint covers 200 square feet. How many cans of paint does Sophia need to buy? Use π ≈ 3.14. Answer: 5 Solution: Calculate the area of the two circular ends. Area of one circle = π × r² = 3.14 × 7² = 3.14 × 49 = 153.86 square feet. Two ends = 2 × 153.86 = 307.72 square feet.
   Full step-by-step solution

   Step 1: Calculate the area of the two circular ends. Area of one circle = π × r² = 3.14 × 7² = 3.14 × 49 = 153.86 square feet. Two ends = 2 × 153.86 = 307.72 square feet. Step 2: Calculate the curved surface area. Curved area = 2 × π × r × h = 2 × 3.14 × 7 × 12 = 2 × 3.14 × 84 = 527.52 square feet. Step 3: Total surface area = 307.72 + 527.52 = 835.24 square feet. Step 4: Divide total area by coverage per can: 835.24 ÷ 200 = 4.1762 cans. Since she can't buy a fraction of a can, she needs to round up to 5 cans. The answer is 5 cans.