Area and Surface Formulas

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Area and Surface Formulas · Grade 7 · Worksheet 3

1. A rectangular prism-shaped shipping container has dimensions 12 meters long, 8 meters wide, and 5 meters high. The company needs to paint the entire exterior surface, including the bottom. What is the total surface area in square meters that needs to be painted? Answer: 392 Solution: Length L = 12 m Width W = 8 m Height H = 5 m We paint the entire exterior surface, including the bottom. That means we paint all 6 faces of the prism.
   Full step-by-step solution

   Let's find the total surface area of the rectangular prism that needs painting. The container has: Length L = 12 m Width W = 8 m Height H = 5 m We paint the entire exterior surface, including the bottom. That means we paint all 6 faces of the prism. Step 1: Recall the formula for total surface area of a rectangular prism: Surface Area = 2 × (L×W + L×H + W×H) Step 2: Compute each of the three types of face areas separately: - Front and back faces: L × H = 12 × 5 = 60 m² each → two of them: 2 × 60 = 120 m² - Left and right faces: W × H = 8 × 5 = 40 m² each → two of them: 2 × 40 = 80 m² - Top and bottom faces: L × W = 12 × 8 = 96 m² each → two of them: 2 × 96 = 192 m² Step 3: Add them up: 120 + 80 + 192 = 392 m² Step 4: Check reasoning: The problem says "including the bottom" — the standard surface area formula already includes bottom and top, so we don’t need to adjust anything. Our calculation includes all 6 faces, so it’s correct. Final answer: 392 square meters.

2. A rectangular prism has length 12 cm, width 8 cm, and height 6 cm. Calculate its surface area. Answer: 432 Solution: Identify the formula for surface area of a rectangular prism: SA = 2lw + 2lh + 2wh Substitute the given dimensions: l = 12 cm, w = 8 cm, h = 6 cm Calculate 2lw = 2 × 12 × 8 = 192 Calculate 2lh = 2 × 12 × 6 = 144 Calculate 2wh = 2 × 8 × 6 = 96 Add all three results: 192 + 144 + 96 = 432 Include…
   Full step-by-step solution

   Step 1: Identify the formula for surface area of a rectangular prism: SA = 2lw + 2lh + 2wh Step 2: Substitute the given dimensions: l = 12 cm, w = 8 cm, h = 6 cm Step 3: Calculate 2lw = 2 × 12 × 8 = 192 Step 4: Calculate 2lh = 2 × 12 × 6 = 144 Step 5: Calculate 2wh = 2 × 8 × 6 = 96 Step 6: Add all three results: 192 + 144 + 96 = 432 Step 7: Include units: 432 cm² The surface area is 432 cm².

3. Emma is designing a new packaging box for her artisanal candles. The box is shaped like a rectangular prism with dimensions 24 cm long, 18 cm wide, and 12 cm high. She needs to calculate the total surface area to determine how much decorative wrapping paper she needs to cover the entire box. What is the total surface area of the box in square centimeters? Answer: 1872 Solution: Identify the dimensions: length = 24 cm, width = 18 cm, height = 12 cm Calculate area of front and back faces: 2 × (length × height) = 2 × (24 × 12) = 2 × 288 = 576 cm² Calculate area of left and right faces: 2 × (width × height) = 2 × (18 × 12) = 2 × 216 = 432 cm² Calculate area of top and…
   Full step-by-step solution

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

4. Mason is designing a cylindrical water tower for a small town. The tower has a radius of 14 meters and a height of 22 meters. He needs to cover the curved side surface and the top circular face with a special reflective coating. The bottom of the tower rests on the ground and does not need coating. What is the total surface area, in square meters, that needs to be coated? Use π = 3.14. Answer: 2549.68 Solution: Identify the dimensions. Radius r = 14 m, height h = 22 m, π = 3.14. Calculate the area of the top circular face.
   Full step-by-step solution

   Step 1: Identify the dimensions. Radius r = 14 m, height h = 22 m, π = 3.14. Step 2: Calculate the area of the top circular face. Area of top = π × r² = 3.14 × (14)² = 3.14 × 196 = 615.44 m² Step 3: Calculate the circumference of the circular base. Circumference = 2 × π × r = 2 × 3.14 × 14 = 6.28 × 14 = 87.92 m Step 4: Calculate the area of the curved side (lateral surface area). Lateral area = circumference × height = 87.92 × 22 = 1934.24 m² Step 5: Add the top area and lateral area to find the total coating area. Total area = 615.44 + 1934.24 = 2549.68 m² The total surface area that needs coating is 2549.68 square meters.