3D Cross Sections

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Answer Key & Explanations

3D Cross Sections · Grade 6 · Worksheet 1

1. Liam slices a cylinder with a radius of 15 cm and a height of 40 cm. He makes a vertical cut straight down through the center of the circular base. What is the area of the rectangular cross section? Answer: 1200 square cm Solution: A vertical cut through the center of a cylinder creates a rectangular cross section. Step 2: The width of the rectangle is the diameter of the circular base. The radius is 15 cm, so the diameter is 2 × 15 = 30 cm.
   Full step-by-step solution

   Step 1: A vertical cut through the center of a cylinder creates a rectangular cross section. Step 2: The width of the rectangle is the diameter of the circular base. The radius is 15 cm, so the diameter is 2 × 15 = 30 cm. Step 3: The height of the rectangle is the height of the cylinder, which is 40 cm. Step 4: Area of a rectangle = width × height = 30 × 40 = 1200 square cm. The answer is 1200 square cm.

2. Liam is designing a new community garden shed shaped like a rectangular prism. The shed measures 4.2 meters long, 3.6 meters wide, and 2.8 meters tall. If Liam draws a horizontal cross-section of the shed at a height of 1.5 meters above the ground, what will be the area of this cross-section in square meters? Answer: 15.12 Solution: A horizontal cross-section parallel to the base of a rectangular prism will always be a rectangle with the same length and width as the base. The length of the cross-section is 4.2 meters.
   Full step-by-step solution

   Step 1: A horizontal cross-section parallel to the base of a rectangular prism will always be a rectangle with the same length and width as the base. Step 2: The length of the cross-section is 4.2 meters. Step 3: The width of the cross-section is 3.6 meters. Step 4: Calculate the area: Area = length × width = 4.2 × 3.6 Step 5: 4.2 × 3.6 = (4.2 × 3) + (4.2 × 0.6) = 12.6 + 2.52 = 15.12 Step 6: The area of the cross-section is 15.12 square meters.

3. Liam is building a rectangular prism-shaped planter box for his garden. The box is 2.4 meters long, 1.5 meters wide, and 0.8 meters high. If Liam makes a vertical cut parallel to the end face of the box, what is the area, in square meters, of the cross-sectional rectangle he creates? Answer: 1.2 Solution: Length = 2.4 m Width = 1.5 m Height = 0.8 m A vertical cut parallel to the end face means the cut is parallel to one of the smaller sides (the width–height face).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the shape and the cut** The box is a rectangular prism with: Length = 2.4 m Width = 1.5 m Height = 0.8 m A vertical cut parallel to the **end face** means the cut is parallel to one of the smaller sides (the width–height face). --- **Step 2: Identify the cross-section** If the cut is parallel to the end face, then the cross-section is a rectangle with the same dimensions as the end face. The end face has: Width = 1.5 m Height = 0.8 m --- **Step 3: Calculate the area of the cross-section** Area = Width × Height Area = 1.5 × 0.8 --- **Step 4: Perform the multiplication** 1.5 × 0.8 = 1.20 --- **Step 5: Final answer** The cross-sectional area is 1.2 square meters. --- **Answer:** 1.2

4. Carlos is designing a new community pool shaped like a rectangular prism. The pool will be 25 meters long, 12 meters wide, and 3 meters deep. If the city planners want to install a safety platform that creates a horizontal cross-section 1.2 meters below the water surface, what will be the area of this rectangular cross-section in square meters? Answer: 300 Solution: Identify the dimensions of the rectangular cross-section. Since this is a horizontal cut through a rectangular prism, the cross-section will have the same length and width as the base. The length of the cross-section is 25 meters.
   Full step-by-step solution

   Step 1: Identify the dimensions of the rectangular cross-section. Since this is a horizontal cut through a rectangular prism, the cross-section will have the same length and width as the base. Step 2: The length of the cross-section is 25 meters. Step 3: The width of the cross-section is 12 meters. Step 4: Calculate the area using the formula: Area = length × width Step 5: Area = 25 × 12 = 300 square meters Step 6: The depth of the cut (1.2 meters) doesn't affect the area since all horizontal cross-sections in a rectangular prism are identical regardless of height. The area of the rectangular cross-section is 300 square meters.

5. Noah is designing a new community swimming pool shaped like a rectangular prism. The pool will be 25 meters long, 12 meters wide, and 3 meters deep. If Noah draws a horizontal cross-section of the pool at a depth of 1.5 meters below the water surface, what will be the area of this cross-section in square meters? Answer: 300 Solution: A horizontal cross-section parallel to the base of a rectangular prism will have the same length and width as the base. The pool has a length of 25 meters and a width of 12 meters.
   Full step-by-step solution

   Step 1: A horizontal cross-section parallel to the base of a rectangular prism will have the same length and width as the base. Step 2: The pool has a length of 25 meters and a width of 12 meters. Step 3: The area of the rectangular cross-section is length × width. Step 4: Calculate: 25 × 12 = 300 Step 5: The area of the cross-section is 300 square meters.