Cross Sections

lessonbunny.com

Answer Key & Explanations

Cross Sections · Grade 7 · Worksheet 2

1. A right rectangular prism has dimensions 14 cm by 9 cm by 22 cm. A plane cuts the prism parallel to the base (the 14 cm by 9 cm face). What is the area of the resulting cross-section? Answer: 126 Solution: Identify the base dimensions. The base is a rectangle with length 14 cm and width 9 cm. When a plane cuts parallel to the base, the cross-section is a rectangle congruent to the base.
   Full step-by-step solution

   Step 1: Identify the base dimensions. The base is a rectangle with length 14 cm and width 9 cm. Step 2: When a plane cuts parallel to the base, the cross-section is a rectangle congruent to the base. Step 3: The area of a rectangle is length times width. Step 4: Area = 14 cm × 9 cm = 126 square cm. The answer is 126.

2. Liam is building a rectangular prism-shaped planter box for his garden. The box has a length of 120 cm, a width of 80 cm, and a height of 60 cm. If Liam makes a vertical cut parallel to the end face of the prism, what is the area of the cross-sectional shape he creates? Answer: 4800 cm² Solution: - Length = 120 cm - Width = 80 cm - Height = 60 cm A vertical cut parallel to the end face means the cut is made along a plane parallel to one of the two smaller faces at the ends of the prism.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangular prism with: - Length = 120 cm - Width = 80 cm - Height = 60 cm A vertical cut parallel to the **end face** means the cut is made along a plane parallel to one of the two smaller faces at the ends of the prism. --- **Step 2: Identify the "end face"** Usually, the "end face" of a rectangular prism is the face with the smallest dimension in one direction. But here, the length (120 cm) is the longest dimension, so the "end faces" are the ones perpendicular to the length. That means the end face has dimensions: **width × height** = 80 cm × 60 cm. --- **Step 3: Visualize the cut** A vertical cut parallel to the end face means we cut along a plane that is parallel to the 80 cm × 60 cm face, and the cut goes vertically (so the height direction is preserved in the cross-section). Since the cut is parallel to the end face, the cross-section will have the same shape and dimensions as the end face: a rectangle with width 80 cm and height 60 cm. --- **Step 4: Calculate the area of the cross-section** Area of rectangle = width × height Area = 80 cm × 60 cm Area = 4800 cm² --- **Step 5: Conclusion** The cross-section is a rectangle of 80 cm by 60 cm, so its area is 4800 cm². --- **Final answer:** 4800 cm²

3. Noah is constructing a large-scale model of a skyscraper for his school's architecture project. The model is shaped like a right rectangular prism with a length of 51 cm, a width of 26 cm, and a height of 91 cm. To study the building's interior layout, Noah makes a vertical cut through the model parallel to the front face (the face with dimensions length and height). What is the area, in square centimeters, of the cross-sectional shape created by this cut? Answer: 4641 Solution: Identify the shape of the cross-section. A vertical cut parallel to the front face of a right rectangular prism creates a rectangle. The front face has length (51 cm) and height (91 cm).
   Full step-by-step solution

   Step 1: Identify the shape of the cross-section. A vertical cut parallel to the front face of a right rectangular prism creates a rectangle. The dimensions of this rectangle are the same as the dimensions of the front face. Step 2: Determine the dimensions of the front face. The front face has length (51 cm) and height (91 cm). So the cross-section rectangle has length = 51 cm and height = 91 cm. Step 3: Calculate the area of the rectangle. Area = length × height = 51 × 91. Step 4: Multiply: 51 × 91 = 51 × (90 + 1) = 51 × 90 + 51 × 1 = 4590 + 51 = 4641. Step 5: Include units: The area is 4641 square centimeters. The answer is 4641.