3D Cross Sections

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Answer Key & Explanations

3D Cross Sections · Grade 6 · Worksheet 3

1. A rectangular prism has a length of 15 cm, a width of 9 cm, and a height of 21 cm. It is sliced horizontally at a height of 7 cm from the base. What is the area of the resulting cross section? Answer: 135 Solution: Identify the shape of the cross section. A horizontal slice through a rectangular prism is parallel to the base, so the cross section is a rectangle with the same length and width as the base. The length of the base is 15 cm and the width is 9 cm.
   Full step-by-step solution

   Step 1: Identify the shape of the cross section. A horizontal slice through a rectangular prism is parallel to the base, so the cross section is a rectangle with the same length and width as the base. Step 2: The length of the base is 15 cm and the width is 9 cm. The height of the cut (7 cm) does not affect the dimensions of the cross section. Step 3: Calculate the area of the rectangular cross section: Area = length × width = 15 cm × 9 cm = 135 square cm. The answer is 135.

2. Emma slices a rectangular prism that is 20 cm long, 15 cm wide, and 10 cm high with a horizontal cut halfway up its height. What is the area of the cross section? Answer: 300 square cm Solution: A horizontal cut through a rectangular prism creates a cross section that is a rectangle parallel to the base. The cross section has the same length and width as the base of the prism. The length is 20 cm and the width is 15 cm.
   Full step-by-step solution

   Step 1: A horizontal cut through a rectangular prism creates a cross section that is a rectangle parallel to the base. Step 2: The cross section has the same length and width as the base of the prism. Step 3: The length is 20 cm and the width is 15 cm. Step 4: Area = length × width = 20 × 15 = 300 square cm. The answer is 300 square cm.

3. (-15) + 28 - (-7) = ? Answer: 20 Solution: (-15) + 28 - (-7) Subtracting a negative is the same as adding a positive: - (-7) = + 7 (-15) + 28 + 7 First: (-15) + 28 Think of it as 28 - 15 = 13 So now we have: 13 + 7 13 + 7 = 20 Final Answer: 20
   Full step-by-step solution

   Let's solve step-by-step: Step 1: Write the problem clearly (-15) + 28 - (-7) Step 2: Interpret the subtraction of a negative number Subtracting a negative is the same as adding a positive: - (-7) = + 7 So the expression becomes: (-15) + 28 + 7 Step 3: Add the numbers from left to right First: (-15) + 28 Think of it as 28 - 15 = 13 So now we have: 13 + 7 Step 4: Add the last number 13 + 7 = 20 Final Answer: 20

4. Maria is designing a new cylindrical water tank for her school's science project. The tank has a diameter of 1.8 meters and a height of 2.5 meters. If she makes a horizontal cut parallel to the base at a height of 1.2 meters from the bottom, what will be the area of the circular cross-section she creates? Use π ≈ 3.14 and round your answer to the nearest hundredth. Answer: 2.54 Solution: Identify the shape of the cross-section. When cutting a cylinder horizontally parallel to the base, the cross-section is always a circle. The diameter of the tank is 1.8 meters, so the radius is half of that: 1.8 ÷ 2 = 0.9 meters.
   Full step-by-step solution

   Step 1: Identify the shape of the cross-section. When cutting a cylinder horizontally parallel to the base, the cross-section is always a circle. Step 2: Determine the radius of the circular cross-section. The diameter of the tank is 1.8 meters, so the radius is half of that: 1.8 ÷ 2 = 0.9 meters. Step 3: Calculate the area of the circular cross-section using the formula A = πr². Step 4: Substitute the values: A = 3.14 × (0.9)² Step 5: Calculate (0.9)² = 0.81 Step 6: Multiply: 3.14 × 0.81 = 2.5434 Step 7: Round to the nearest hundredth: 2.54 Step 8: Include units: The area is 2.54 square meters. The answer is 2.54.

5. Liam is building a rectangular prism-shaped planter box for his garden that is 2.4 meters long, 1.5 meters wide, and 0.8 meters high. He wants to visualize the different cross-sectional shapes he could see if he sliced the box. If Liam makes a horizontal cut parallel to the base at a height of 0.3 meters, what is the area of the rectangular cross-section he would see? Answer: 3.6 Solution: - Length = 2.4 m - Width = 1.5 m - Height = 0.8 m A horizontal cut parallel to the base means the cross-section is parallel to the length and width dimensions. The height of the cut is 0.3 m from the base.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangular prism with dimensions: - Length = 2.4 m - Width = 1.5 m - Height = 0.8 m A horizontal cut parallel to the base means the cross-section is parallel to the length and width dimensions. The height of the cut is 0.3 m from the base. --- **Step 2: Determine cross-section dimensions** A horizontal cross-section at any height has the same length and width as the base, because the sides are vertical. - Length of cross-section = 2.4 m - Width of cross-section = 1.5 m The height of the cut (0.3 m) doesn't change the length and width, since the box is a right prism with vertical sides. --- **Step 3: Calculate area of cross-section** Area of rectangle = length × width Area = 2.4 × 1.5 --- **Step 4: Perform multiplication** First, 2.4 × 1.5: 2.4 × 1 = 2.4 2.4 × 0.5 = 1.2 Add: 2.4 + 1.2 = 3.6 --- **Step 5: Final answer** The area of the rectangular cross-section is 3.6 square meters. --- **Answer:** 3.6