Area and Surface Formulas

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Answer Key & Explanations

Area and Surface Formulas · Grade 7 · Worksheet 2

1. Aroha is painting a cylindrical water tank for a community art project. The tank has a radius of 1.5 meters and a height of 3.2 meters. She needs to paint the curved side and the top circular lid, but not the bottom. What is the total surface area she needs to paint in square meters? Use π ≈ 3.14 and round your answer to the nearest tenth. Answer: 37.2 Solution: Find the area of the curved side (lateral surface area). The formula is 2πrh, where r is the radius and h is the height. 2 × 3.14 × 1.5 × 3.2 = 2 × 3.14 × 4.8 = 30.144 square meters.
   Full step-by-step solution

   Step 1: Find the area of the curved side (lateral surface area). The formula is 2πrh, where r is the radius and h is the height. 2 × 3.14 × 1.5 × 3.2 = 2 × 3.14 × 4.8 = 30.144 square meters. Step 2: Find the area of the top circular lid. The formula is πr^2. 3.14 × (1.5 × 1.5) = 3.14 × 2.25 = 7.065 square meters. Step 3: Add the two areas together. 30.144 + 7.065 = 37.209 square meters. Step 4: Round to the nearest tenth. 37.209 rounds to 37.2. The total surface area Aroha needs to paint is 37.2 square meters.

2. A cylinder has a radius of 5 cm and a height of 20 cm. Calculate its total surface area. (Use π = 3.14) Answer: 785 Solution: Calculate the area of one circular base: πr² = 3.14 × 5² = 3.14 × 25 = 78.5 cm². Two bases: 2 × 78.5 = 157 cm².
   Full step-by-step solution

   Step 1: Calculate the area of one circular base: πr² = 3.14 × 5² = 3.14 × 25 = 78.5 cm². Two bases: 2 × 78.5 = 157 cm². Step 2: Calculate the curved surface area: circumference × height = 2πr × h = 2 × 3.14 × 5 × 20 = 6.28 × 100 = 628 cm². Step 3: Add the areas: 157 + 628 = 785 cm². The total surface area is 785 cm².

3. A rectangular prism has dimensions 24 cm × 18 cm × 12 cm. Calculate its surface area. Answer: 1872 Solution: Identify the three dimensions: length = 24 cm, width = 18 cm, height = 12 cm Calculate area of front/back faces: 2 × (length × height) = 2 × (24 × 12) = 2 × 288 = 576 cm² Calculate area of left/right faces: 2 × (width × height) = 2 × (18 × 12) = 2 × 216 = 432 cm² Calculate area of top/bottom…
   Full step-by-step solution

   Step 1: Identify the three dimensions: length = 24 cm, width = 18 cm, height = 12 cm Step 2: Calculate area of front/back faces: 2 × (length × height) = 2 × (24 × 12) = 2 × 288 = 576 cm² Step 3: Calculate area of left/right faces: 2 × (width × height) = 2 × (18 × 12) = 2 × 216 = 432 cm² Step 4: Calculate area of top/bottom faces: 2 × (length × width) = 2 × (24 × 18) = 2 × 432 = 864 cm² Step 5: Add all areas: 576 + 432 + 864 = 1872 cm² The surface area is 1872 cm².

4. 2 × (12 × 8 + 12 × 5 + 8 × 5) = ? Answer: 392 Solution: Calculate the three products inside the parentheses: 12 × 8 = 96, 12 × 5 = 60, and 8 × 5 = 40 Add these three products: 96 + 60 + 40 = 196 Multiply the sum by 2: 2 × 196 = 392 The answer is 392.
   Full step-by-step solution

   Step 1: Calculate the three products inside the parentheses: 12 × 8 = 96, 12 × 5 = 60, and 8 × 5 = 40 Step 2: Add these three products: 96 + 60 + 40 = 196 Step 3: Multiply the sum by 2: 2 × 196 = 392 The answer is 392.