Cross Sections

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Cross Sections · Grade 7 · Worksheet 2

1. 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.

2. Liam is designing a custom chocolate bar shaped like a right triangular prism. The triangular base has a height of 8 cm and a base length of 6 cm. The chocolate bar is 20 cm long. If Liam cuts the chocolate bar with a plane parallel to its rectangular faces, what is the area of the cross-section that is a rectangle measuring 20 cm by 8 cm? Answer: 160 square centimeters Solution: The chocolate bar is a right triangular prism. - height = 8 cm - base length = 6 cm The prism length = 20 cm (this is the distance between the two triangular faces).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the shape** The chocolate bar is a right triangular prism. The triangular base has: - height = 8 cm - base length = 6 cm The prism length = 20 cm (this is the distance between the two triangular faces). --- **Step 2: Visualize the cross-section** The problem says Liam cuts the chocolate bar with a plane **parallel to its rectangular faces**. A right triangular prism has 5 faces: - 2 triangular faces (front and back) - 3 rectangular faces (side faces) The rectangular faces are: 1. One rectangle of size 20 cm × 8 cm (this is the face corresponding to the height of the triangle) 2. One rectangle of size 20 cm × 6 cm (this is the face corresponding to the base of the triangle) 3. One rectangle of size 20 cm × hypotenuse (not needed here) --- **Step 3: Interpret the cutting plane** A plane parallel to the rectangular faces means the cut is perpendicular to the triangular faces. But more specifically, if the plane is parallel to the **rectangular face that is 20 cm by 8 cm**, then the cross-section will be a rectangle of the same dimensions as that face. --- **Step 4: Identify the correct cross-section** The problem says: "cross-section that is a rectangle measuring 20 cm by 8 cm". This means the cut is made parallel to the triangular base's height direction, so the cross-section is a rectangle with: - length = prism length = 20 cm - width = height of triangle = 8 cm --- **Step 5: Calculate the area of the cross-section** Area of rectangle = length × width Area = 20 cm × 8 cm = 160 cm² --- **Step 6: Conclusion** The cross-section is simply the same as one of the prism's rectangular faces, so no further calculation is needed. --- **Final Answer:** 160 square centimeters

3. A construction company is building a concrete support beam shaped like a right prism with a trapezoidal base. The trapezoid has parallel sides measuring 2.5 meters and 4.5 meters, with a height of 1.5 meters. The beam is 12 meters long. If workers make a vertical cut perpendicular to the length of the beam, what is the area of the resulting cross-section in square meters? Answer: 5.25 Solution: Identify that a vertical cut perpendicular to the length of a prism reveals the base shape. The cross-section is a trapezoid with parallel sides of 2.5 m and 4.5 m, and height of 1.5 m.
   Full step-by-step solution

   Step 1: Identify that a vertical cut perpendicular to the length of a prism reveals the base shape. Step 2: The cross-section is a trapezoid with parallel sides of 2.5 m and 4.5 m, and height of 1.5 m. Step 3: Use the trapezoid area formula: Area = (1/2) × (base1 + base2) × height Step 4: Substitute the values: Area = (1/2) × (2.5 + 4.5) × 1.5 Step 5: Calculate inside parentheses: 2.5 + 4.5 = 7 Step 6: Multiply: (1/2) × 7 = 3.5 Step 7: Multiply by height: 3.5 × 1.5 = 5.25 Step 8: The area of the cross-section is 5.25 square meters.

4. A right rectangular prism has dimensions 18 cm by 12 cm by 24 cm. A slice is made parallel to the base. What is the area of the cross-section? Answer: 216 Solution: Identify the base of the prism. The base is the face with dimensions 18 cm by 12 cm. A slice parallel to the base creates a cross-section that is identical in shape and size to the base.
   Full step-by-step solution

   Step 1: Identify the base of the prism. The base is the face with dimensions 18 cm by 12 cm. Step 2: A slice parallel to the base creates a cross-section that is identical in shape and size to the base. Step 3: The cross-section is a rectangle with length 18 cm and width 12 cm. Step 4: Calculate the area: Area = length × width = 18 cm × 12 cm = 216 square cm. The answer is 216.

5. A right rectangular prism has a base with dimensions 20 cm by 15 cm and a height of 30 cm. A plane cuts the prism parallel to the base at a height of 10 cm from the base. What is the area of the cross-section? Answer: 300 Solution: When a right rectangular prism is cut by a plane parallel to the base, the cross-section is a rectangle with the same dimensions as the base. The base dimensions are 20 cm by 15 cm.
   Full step-by-step solution

   Step 1: When a right rectangular prism is cut by a plane parallel to the base, the cross-section is a rectangle with the same dimensions as the base. Step 2: The base dimensions are 20 cm by 15 cm. Step 3: The cross-section is a rectangle with length 20 cm and width 15 cm. Step 4: Area of a rectangle = length × width = 20 × 15 = 300. Step 5: The area of the cross-section is 300 square centimeters. The answer is 300.