Cross Sections

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Cross Sections · Grade 7 · Worksheet 1

1. A right rectangular prism has dimensions 27 cm by 12 cm by 17 cm. Mason makes a slice parallel to the base. What is the area of the cross-section? Answer: 324 Solution: Identify the base of the prism. The base is a rectangle with dimensions 27 cm by 12 cm. A slice parallel to the base creates a cross-section that is congruent to the base.
   Full step-by-step solution

   Step 1: Identify the base of the prism. The base is a rectangle with dimensions 27 cm by 12 cm. Step 2: A slice parallel to the base creates a cross-section that is congruent to the base. Step 3: The cross-section is a rectangle with the same length and width as the base: 27 cm and 12 cm. Step 4: Calculate the area: Area = length × width = 27 × 12 = 324. The answer is 324.

2. Mason is building a decorative concrete stepping stone shaped like a right rectangular prism. The stone measures 32 cm in length, 22 cm in width, and 7 cm in height. He wants to create a design by making a horizontal cut parallel to the base, exactly halfway up the height of the prism. What is the area, in square centimeters, of the cross-section created by this cut? Answer: 704 Solution: Identify the shape of the cross-section. A horizontal cut parallel to the base of a right rectangular prism creates a rectangle that is identical in shape and size to the base. The length and width of the base are the same as the prism's length and width: 32 cm and 22 cm.
   Full step-by-step solution

   Step 1: Identify the shape of the cross-section. A horizontal cut parallel to the base of a right rectangular prism creates a rectangle that is identical in shape and size to the base. Step 2: Determine the dimensions of the cross-section. The length and width of the base are the same as the prism's length and width: 32 cm and 22 cm. The height of the cut (halfway at 3.5 cm) does not affect the area of the cross-section. Step 3: Calculate the area of the rectangle. Area = length x width = 32 cm x 22 cm. Step 4: Multiply: 32 x 22 = 704. Step 5: Include the units: 704 square centimeters. The answer is 704.

3. Maya is designing a custom aquarium shaped like a right triangular prism for a marine exhibit. The triangular base is a right triangle with legs measuring 12 cm and 16 cm. The aquarium is 150 cm long. If Maya makes a vertical cut through the aquarium parallel to the triangular ends, what is the area of the resulting triangular cross-section in square centimeters? Answer: 96 Solution: Identify the cross-section shape. When cutting a prism parallel to its bases, the cross-section is identical to the base shape. Recall the formula for the area of a right triangle: Area = (1/2) × base × height The triangle has legs measuring 12 cm and 16 cm.
   Full step-by-step solution

   Step 1: Identify the cross-section shape. When cutting a prism parallel to its bases, the cross-section is identical to the base shape. In this case, the base is a right triangle. Step 2: Recall the formula for the area of a right triangle: Area = (1/2) × base × height Step 3: The triangle has legs measuring 12 cm and 16 cm. These serve as the base and height of the triangle. Step 4: Calculate the area: Area = (1/2) × 12 cm × 16 cm = (1/2) × 192 cm² = 96 cm² Step 5: The length of the prism (150 cm) is not needed since we're finding the area of the cross-section parallel to the triangular ends. The area of the triangular cross-section is 96 square centimeters.

4. A right rectangular prism has dimensions 18 cm by 13 cm by 22 cm. A slice is made parallel to the base. What is the area of the cross-section? Answer: 234 Solution: Identify the base dimensions. The base of the prism is a rectangle with length 18 cm and width 13 cm. When slicing parallel to the base, the cross-section is a rectangle with the same length and width as the base.
   Full step-by-step solution

   Step 1: Identify the base dimensions. The base of the prism is a rectangle with length 18 cm and width 13 cm. Step 2: When slicing parallel to the base, the cross-section is a rectangle with the same length and width as the base. Step 3: Calculate the area of the cross-section: Area = length × width = 18 cm × 13 cm = 234 square cm. The answer is 234.

5. (3² × 4) - (15 ÷ 3) = ? Answer: 31 Solution: We have: (3² × 4) - (15 ÷ 3) 3² means 3 × 3 = 9 So now we have: (9 × 4) - (15 ÷ 3) Perform the multiplication inside the first parentheses 9 × 4 = 36 Now we have: 36 - (15 ÷ 3) Perform the division inside the second parentheses 15 ÷ 3 = 5 Now we have: 36 - 5 36 - 5 = 31 Final Answer: 31
   Full step-by-step solution

   Let's solve step-by-step. We have: (3² × 4) - (15 ÷ 3) **Step 1: Handle the exponent inside the first parentheses** 3² means 3 × 3 = 9 So now we have: (9 × 4) - (15 ÷ 3) **Step 2: Perform the multiplication inside the first parentheses** 9 × 4 = 36 Now we have: 36 - (15 ÷ 3) **Step 3: Perform the division inside the second parentheses** 15 ÷ 3 = 5 Now we have: 36 - 5 **Step 4: Perform the subtraction** 36 - 5 = 31 **Final Answer:** 31