Cross Sections

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Cross Sections · Grade 7 · Worksheet 1

1. A right rectangular prism has dimensions 21 cm × 15 cm × 25 cm. A slice is made parallel to the base. What is the shape and area of the cross-section? Answer: 315 square centimeters Solution: Identify the base of the prism. The base is the face with dimensions 21 cm by 15 cm (the first two numbers given).
   Full step-by-step solution

   Step 1: Identify the base of the prism. The base is the face with dimensions 21 cm by 15 cm (the first two numbers given). Step 2: When a slice is made parallel to the base, the cross-section is a rectangle with the same length and width as the base. Step 3: The cross-section has dimensions 21 cm by 15 cm. Step 4: Calculate the area: Area = length × width = 21 × 15 = 315 square centimeters. The answer is 315 square centimeters.

2. Isabella is creating a large-scale art installation for a science museum. She builds a right rectangular prism structure that is 27 feet long, 12 feet wide, and 7 feet tall. For a special lighting effect, she needs to make a vertical cut through the prism parallel to the end face (the face that is 12 feet by 7 feet). What is the area, in square feet, of the cross-sectional shape created by this cut? Answer: 84 Solution: Identify the cross-section shape. A vertical cut parallel to the end face creates a rectangle that has the same dimensions as that end face. The end face of the prism measures 12 feet in width and 7 feet in height.
   Full step-by-step solution

   Step 1: Identify the cross-section shape. A vertical cut parallel to the end face creates a rectangle that has the same dimensions as that end face. Step 2: The end face of the prism measures 12 feet in width and 7 feet in height. Step 3: Calculate the area of the rectangle: Area = width x height = 12 x 7 = 84. Step 4: Include units: The area is 84 square feet. The answer is 84.

3. A construction company is building a concrete ramp in the shape of a right triangular prism. The ramp's triangular base has a height of 2.5 meters and a base length of 6 meters. If the ramp extends 15 meters in length, what is the total volume of concrete needed for the ramp? Answer: 112.5 Solution: The ramp is a right triangular prism. This means the cross-section is a triangle, and it extends in length. - Height = 2.5 meters - Base length = 6 meters Calculate the area of the triangular base.
   Full step-by-step solution

   Step 1: Understand the shape of the ramp. The ramp is a right triangular prism. This means the cross-section is a triangle, and it extends in length. Step 2: Identify the dimensions of the triangular base. The triangular base has: - Height = 2.5 meters - Base length = 6 meters Step 3: Calculate the area of the triangular base. The area of a triangle is (1/2) * base * height. Area = (1/2) * 6 * 2.5 First, 1/2 * 6 = 3 Then, 3 * 2.5 = 7.5 So, the area of the triangular base is 7.5 square meters. Step 4: Identify the length of the prism. The ramp extends 15 meters in length. This is the length of the prism. Step 5: Calculate the volume of the prism. The volume of a prism is the area of the base times the length of the prism. Volume = base area * length Volume = 7.5 * 15 Step 6: Perform the multiplication. 7.5 * 15 = 112.5 Step 7: State the final answer. The total volume of concrete needed is 112.5 cubic meters.

4. Liam is designing a custom chocolate bar shaped like a right triangular prism. The triangular face has a base of 6 cm and a height of 4 cm. If the chocolate bar is 20 cm long, what is the total volume of chocolate in cubic centimeters? Answer: 240 Solution: It is a right triangular prism, which means it has a triangular face extended along a length. The volume is found by multiplying the area of the triangular face by the length of the prism. The triangular face has a base of 6 cm and a height of 4 cm.
   Full step-by-step solution

   Step 1: Understand the shape of the chocolate bar. It is a right triangular prism, which means it has a triangular face extended along a length. The volume is found by multiplying the area of the triangular face by the length of the prism. Step 2: Find the area of the triangular face. The triangular face has a base of 6 cm and a height of 4 cm. The formula for the area of a triangle is: Area = (base * height) / 2 Substitute the values: Area = (6 * 4) / 2 Area = 24 / 2 Area = 12 square cm. Step 3: Find the volume of the prism. The volume of a prism is: Volume = (Area of the base) * (Length of the prism) Here, the base is the triangular face, so: Volume = 12 * 20 Volume = 240 cubic cm. Step 4: State the final answer. The total volume of chocolate is 240 cubic cm.

5. Isabella is an architect designing a modern art museum. The museum's main gallery is shaped like a right rectangular prism with dimensions 28 meters in length, 16 meters in width, and 12 meters in height. For a special exhibit, she plans to install a large vertical partition wall that cuts through the gallery parallel to the front face (the face that is 28 meters wide and 12 meters tall). What is the area, in square meters, of the cross-sectional shape created by this cut? Answer: 336 Solution: Identify the cross-section shape. A vertical cut parallel to the front face creates a rectangle that has the same dimensions as the front face. Step 2: Determine the dimensions of this rectangle.
   Full step-by-step solution

   Step 1: Identify the cross-section shape. A vertical cut parallel to the front face creates a rectangle that has the same dimensions as the front face. Step 2: Determine the dimensions of this rectangle. The front face has a width of 28 meters and a height of 12 meters. Step 3: Calculate the area of the rectangle. Area = width x height = 28 x 12. Step 4: Multiply: 28 x 12 = 336. Step 5: Include units: The area is 336 square meters. The answer is 336.

6. (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