Cross Sections

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Cross Sections · Grade 7 · Worksheet 3

1. A rectangular prism has a length of 15 cm, width of 8 cm, and height of 12 cm. If you make a vertical cut parallel to the face with dimensions 8 cm by 12 cm, what is the area of the cross-section created by this cut? Answer: 96 Solution: - Length = 15 cm - Width = 8 cm - Height = 12 cm - Front face: width × height = 8 cm × 12 cm - Side face: length × height = 15 cm × 12 cm - Top face: length × width = 15 cm × 8 cm The problem says: "a vertical cut parallel to the face with dimensions 8 cm by 12 cm." - The face with dimensions 8…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the shape and the cut** We have a rectangular prism with: - Length = 15 cm - Width = 8 cm - Height = 12 cm The faces are: - Front face: width × height = 8 cm × 12 cm - Side face: length × height = 15 cm × 12 cm - Top face: length × width = 15 cm × 8 cm --- **Step 2: Interpret the cut** The problem says: "a vertical cut parallel to the face with dimensions 8 cm by 12 cm." - The face with dimensions 8 cm by 12 cm is the front face (width × height). - "Vertical" here means the cut is perpendicular to the base (so it goes up/down). - "Parallel to the front face" means the cut is made such that the cross-section we see is parallel to the front face. If we cut parallel to the front face, we are slicing along a plane parallel to the 8 cm × 12 cm face. That means the cross-section will have the same dimensions as that face: 8 cm by 12 cm. --- **Step 3: Determine the cross-section area** The cross-section is a rectangle with width = 8 cm and height = 12 cm. Area = width × height Area = 8 × 12 Area = 96 cm² --- **Step 4: Conclusion** The area of the cross-section created by the cut is 96 cm². --- **Final Answer:** 96

2. A right rectangular prism has dimensions 20 cm by 15 cm by 25 cm. A plane cuts the prism parallel to its base. What is the shape and area of the cross-section? Answer: 300 square centimeters Solution: Identify the base of the prism. The base is a rectangle with dimensions 20 cm by 15 cm.
   Full step-by-step solution

   Step 1: Identify the base of the prism. The base is a rectangle with dimensions 20 cm by 15 cm. Step 2: When a plane cuts parallel to the base, the cross-section is congruent to the base, so it is also a rectangle with dimensions 20 cm by 15 cm. Step 3: Calculate the area of the rectangle: Area = length × width = 20 cm × 15 cm = 300 square centimeters. The answer is 300 square centimeters.

3. A construction company is building a concrete support column in the shape of a right prism with a trapezoidal base. The trapezoid has bases measuring 2.5 meters and 1.5 meters, with a height of 1 meter. The column is 8 meters tall. If workers make a horizontal cut through the column at any height, what is the area of the cross-section they create? Answer: 2 square meters Solution: Identify that a horizontal cut through a prism parallel to its base creates a cross-section identical to the base shape. The base is a trapezoid with bases of 2.5 m and 1.5 m, and height of 1 m.
   Full step-by-step solution

   Step 1: Identify that a horizontal cut through a prism parallel to its base creates a cross-section identical to the base shape. Step 2: The base is a trapezoid with bases of 2.5 m and 1.5 m, and height of 1 m. Step 3: Calculate the area of the trapezoidal base using the formula: Area = (1/2) × (base1 + base2) × height Step 4: Area = (1/2) × (2.5 + 1.5) × 1 Step 5: Area = (1/2) × 4 × 1 Step 6: Area = 2 × 1 = 2 Step 7: The cross-sectional area is 2 square meters.

4. A right rectangular prism has dimensions 16 cm by 10 cm by 22 cm. A slice is made parallel to the base. What is the shape and area of the cross-section? Answer: rectangle, 160 cm² Solution: The base of a right rectangular prism is a rectangle. Its dimensions are the length and width of the prism. The base dimensions are 16 cm and 10 cm.
   Full step-by-step solution

   Step 1: The base of a right rectangular prism is a rectangle. Its dimensions are the length and width of the prism. Step 2: The base dimensions are 16 cm and 10 cm. Step 3: A slice parallel to the base creates a cross-section that is congruent to the base, so it is also a rectangle with dimensions 16 cm by 10 cm. Step 4: Area of the rectangle = length × width = 16 × 10 = 160 cm². The answer is rectangle, 160 cm².