Cross Sections

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Answer Key & Explanations

Cross Sections · Grade 7 · Worksheet 3

1. A right rectangular prism has a length of 12 cm, width of 8 cm, and height of 15 cm. If you make a vertical cut parallel to the height, what is the area of the cross-section that results in the largest possible rectangle? Answer: 180 Solution: - length = 12 cm - width = 8 cm - height = 15 cm A vertical cut parallel to the height means the cut goes up and down (along the height direction).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a right rectangular prism with: - length = 12 cm - width = 8 cm - height = 15 cm A vertical cut parallel to the height means the cut goes up and down (along the height direction). The cross-section will be a rectangle whose one side is the height (15 cm) and the other side is one of the prism's other dimensions (length or width), depending on the orientation of the cut. --- **Step 2: Identify possible cross-sections from a vertical cut** A vertical cut means the cross-section is a plane containing the vertical line (height = 15 cm). Two obvious cases: 1. **Cut parallel to the length-width face such that the cross-section's horizontal side is the length (12 cm)** Then cross-section rectangle dimensions: 12 cm (horizontal) × 15 cm (vertical) Area = 12 × 15 = 180 cm² 2. **Cut parallel to the width-length face such that the cross-section's horizontal side is the width (8 cm)** Then cross-section rectangle dimensions: 8 cm (horizontal) × 15 cm (vertical) Area = 8 × 15 = 120 cm² --- **Step 3: Check if any other orientation gives a larger rectangle** If we cut vertically but at an angle through the prism, the cross-section is still a rectangle with one side = height (15 cm) and the other side = the length of the intersection of the cutting plane with the base. The base is a 12 cm × 8 cm rectangle. The intersection of a vertical plane with the base is a straight line segment across the base. The length of that segment is maximized when it is a diagonal of the base. Diagonal of base = sqrt(12² + 8²) = sqrt(144 + 64) = sqrt(208) ≈ 14.422 cm. Then cross-section rectangle dimensions: ~14.422 cm × 15 cm Area ≈ 14.422 × 15 ≈ 216.33 cm². --- **Step 4: But is that allowed by the problem's wording?** The problem says: "make a vertical cut parallel to the height" — usually this means the cut is parallel to a lateral face (so the cross-section's horizontal dimension is either the length or the width, not a diagonal). If they meant any vertical plane, then the maximum cross-section rectangle area would be 15 × diagonal of base ≈ 216.33 cm². But the correct answer given is 180, so they must mean the cut is parallel to a pair of opposite faces (so the cross-section is a face-like rectangle: either 12×15 or 8×15). --- **Step 5: Choose the larger of the two possible cross-sections from standard vertical cuts** - Cut parallel to width: cross-section = 8 × 15 = 120 - Cut parallel to length: cross-section = 12 × 15 = 180 Largest = 180. --- **Step 6: Conclusion** The largest possible rectangular cross-section from a vertical cut parallel to height (and thus parallel to a lateral face) is 180 cm². --- **Final answer:** 180

2. A right rectangular prism has dimensions 13 cm by 17 cm by 19 cm. A plane cuts the prism parallel to its base. What is the shape of the cross-section? Answer: rectangle Solution: The base of a right rectangular prism is a rectangle. Step 2: When a plane cuts the prism parallel to the base, the cross-section is identical in shape to the base. Step 3: Therefore, the cross-section is a rectangle.
   Full step-by-step solution

   Step 1: The base of a right rectangular prism is a rectangle. Step 2: When a plane cuts the prism parallel to the base, the cross-section is identical in shape to the base. Step 3: Therefore, the cross-section is a rectangle. The answer is rectangle.

3. A right rectangular prism has dimensions 24 cm by 16 cm by 20 cm. A slice is made parallel to the base, cutting through the prism at a height of 12 cm from the base. What is the area of the cross-section? Answer: 384 Solution: The base of the prism is a rectangle with dimensions 24 cm by 16 cm. A slice parallel to the base creates a cross-section that is congruent to the base. Area of the cross-section = length × width = 24 × 16.
   Full step-by-step solution

   Step 1: The base of the prism is a rectangle with dimensions 24 cm by 16 cm. Step 2: A slice parallel to the base creates a cross-section that is congruent to the base. So the cross-section is also a rectangle with length 24 cm and width 16 cm. Step 3: Area of the cross-section = length × width = 24 × 16. Step 4: 24 × 16 = 384. The area of the cross-section is 384 square centimeters.

4. A right rectangular prism has a base with dimensions 11 cm by 17 cm and a height of 23 cm. If the prism is sliced parallel to its base, what is the area of the resulting cross-section? Answer: 187 square cm Solution: Identify the shape of the cross-section. When a right rectangular prism is sliced parallel to its base, the cross-section is a rectangle that is congruent to the base. The base dimensions are 11 cm and 17 cm.
   Full step-by-step solution

   Step 1: Identify the shape of the cross-section. When a right rectangular prism is sliced parallel to its base, the cross-section is a rectangle that is congruent to the base. Step 2: The base dimensions are 11 cm and 17 cm. Step 3: Calculate the area of the rectangle: Area = length × width = 11 × 17 = 187 square cm. The answer is 187 square cm.