Examples of cross-sections formed by cutting 3D shapes in different orientations.
A diagram illustrating how to cut a 3D shape (cylinder, cone, prism) parallel, perpendicular, or diagonally to its base, showing resulting cross-sections.
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Step-by-step solution for: Cross-Sections of 3-D shapes Worksheet by Math Maker worksheets library
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Show Answer Key & Explanations
Step-by-step solution for: Cross-Sections of 3-D shapes Worksheet by Math Maker worksheets library
Based on the pattern shown in the "Examples" table, here is how to complete the missing parts for the prism row.
1. Analyze the Pattern:
* The table shows three types of 3D shapes: a cylinder, a cone, and a prism (implied by the last row).
* For each shape, it demonstrates what the cross-section (the flat surface created by the cut) looks like when cut in three specific ways:
* Cut parallel to base: The cut is horizontal. The resulting shape is the same as the base.
* Cut perpendicular to base: The cut is vertical (straight down through the center). The resulting shape is usually a rectangle or a triangle depending on the object.
* Cut diagonally: The cut is at a slant/angle. The resulting shape is an ellipse (oval) or a more complex curved/triangular shape.
2. Apply to the Prism Row:
* Original Shape: A triangular prism (a tube with a triangle at both ends).
* Task 1: Cut parallel to base.
* If you slice a triangular prism horizontally (parallel to the triangular base), the cross-section will be identical to the base.
* *Result:* A triangle.
* Task 2: Cut perpendicular to base.
* If you slice a triangular prism vertically (straight down from top to bottom, perpendicular to the base), you are cutting through the rectangular faces.
* *Result:* A rectangle.
* Task 3: Cut diagonally.
* If you slice a triangular prism at an angle (diagonally across the corners), the cross-section creates a shape that connects the sides at different heights.
* *Result:* A trapezoid (a four-sided shape with one pair of parallel sides) or potentially a triangle if cut through just one corner, but typically in these standard geometry problems, a diagonal cut across the main body of a prism results in a trapezoid or a parallelogram depending on the exact angle. Looking at the cylinder example (ellipse) and cone example (hyperbola/parabola section), the prism diagonal cut is most commonly represented as a trapezoid in this context because it cuts through two parallel rectangular faces and the two triangular bases at an angle. However, simpler curriculum often accepts a triangle if the cut goes through a vertex, or a quadrilateral. Let's look closer at the cylinder/cone logic.
* Cylinder diagonal -> Ellipse.
* Cone diagonal -> Curved shape (conic section).
* Prism diagonal -> If you cut off a corner, it's a triangle. If you cut through the middle diagonally, it's a trapezoid or parallelogram. Given the simplicity of the other answers (Triangle for parallel, Rectangle for perpendicular), the diagonal cut is most likely intended to show a Trapezoid or simply a slanted rectangle/parallelogram. In many middle-school contexts, a diagonal cut of a triangular prism is often depicted as a trapezoid.
* *Self-Correction/Refinement:* Let's re-evaluate the "Cut Diagonally" for a triangular prism based on standard educational diagrams.
* Parallel to base: Triangle.
* Perpendicular to base: Rectangle.
* Diagonal: If you slice from one edge to the opposite face, you get a trapezoid. If you slice through just one vertex and the opposite side, you might get a triangle. However, the most distinct "diagonal" cross-section that differs from the others is a trapezoid.
* *Alternative Interpretation:* Sometimes "diagonal" for a prism refers to cutting through the vertices to create a larger triangle inside? No, that's not a cross-section. The cross-section is the face exposed.
* Let's stick to the most common geometric cross-sections for a triangular prism:
1. Parallel to base: Triangle
2. Perpendicular to base: Rectangle
3. Diagonal: Trapezoid (or sometimes Parallelogram)
* *Wait, looking at the incomplete image provided in the prompt's `Final Answer` slot requirement:* The user wants me to solve the problem. The problem is filling in the blanks for the prism row.
* Blank 1 (Parallel): Triangle
* Blank 2 (Perpendicular): Rectangle
* Blank 3 (Diagonal): Trapezoid
Let's double-check the "Diagonal" cut for a triangular prism. If you cut diagonally across the entire length, intersecting all 5 faces, it is a pentagon? No, usually these problems assume simple cuts.
- Cut parallel to base: Triangle.
- Cut perpendicular to base (vertical): Rectangle.
- Cut diagonally (slanted): This usually produces a Trapezoid.
Therefore, the shapes to fill in are:
1. Under "Cut parallel to base": Triangle
2. Under "Cut perpendicular to base": Rectangle
3. Under "Cut diagonally": Trapezoid
Final Answer:
The missing cross-sections for the prism are:
1. Cut parallel to base: Triangle
2. Cut perpendicular to base: Rectangle
3. Cut diagonally: Trapezoid
Step-by-Step Deduction:
1. Analyze the Pattern:
* The table shows three types of 3D shapes: a cylinder, a cone, and a prism (implied by the last row).
* For each shape, it demonstrates what the cross-section (the flat surface created by the cut) looks like when cut in three specific ways:
* Cut parallel to base: The cut is horizontal. The resulting shape is the same as the base.
* Cut perpendicular to base: The cut is vertical (straight down through the center). The resulting shape is usually a rectangle or a triangle depending on the object.
* Cut diagonally: The cut is at a slant/angle. The resulting shape is an ellipse (oval) or a more complex curved/triangular shape.
2. Apply to the Prism Row:
* Original Shape: A triangular prism (a tube with a triangle at both ends).
* Task 1: Cut parallel to base.
* If you slice a triangular prism horizontally (parallel to the triangular base), the cross-section will be identical to the base.
* *Result:* A triangle.
* Task 2: Cut perpendicular to base.
* If you slice a triangular prism vertically (straight down from top to bottom, perpendicular to the base), you are cutting through the rectangular faces.
* *Result:* A rectangle.
* Task 3: Cut diagonally.
* If you slice a triangular prism at an angle (diagonally across the corners), the cross-section creates a shape that connects the sides at different heights.
* *Result:* A trapezoid (a four-sided shape with one pair of parallel sides) or potentially a triangle if cut through just one corner, but typically in these standard geometry problems, a diagonal cut across the main body of a prism results in a trapezoid or a parallelogram depending on the exact angle. Looking at the cylinder example (ellipse) and cone example (hyperbola/parabola section), the prism diagonal cut is most commonly represented as a trapezoid in this context because it cuts through two parallel rectangular faces and the two triangular bases at an angle. However, simpler curriculum often accepts a triangle if the cut goes through a vertex, or a quadrilateral. Let's look closer at the cylinder/cone logic.
* Cylinder diagonal -> Ellipse.
* Cone diagonal -> Curved shape (conic section).
* Prism diagonal -> If you cut off a corner, it's a triangle. If you cut through the middle diagonally, it's a trapezoid or parallelogram. Given the simplicity of the other answers (Triangle for parallel, Rectangle for perpendicular), the diagonal cut is most likely intended to show a Trapezoid or simply a slanted rectangle/parallelogram. In many middle-school contexts, a diagonal cut of a triangular prism is often depicted as a trapezoid.
* *Self-Correction/Refinement:* Let's re-evaluate the "Cut Diagonally" for a triangular prism based on standard educational diagrams.
* Parallel to base: Triangle.
* Perpendicular to base: Rectangle.
* Diagonal: If you slice from one edge to the opposite face, you get a trapezoid. If you slice through just one vertex and the opposite side, you might get a triangle. However, the most distinct "diagonal" cross-section that differs from the others is a trapezoid.
* *Alternative Interpretation:* Sometimes "diagonal" for a prism refers to cutting through the vertices to create a larger triangle inside? No, that's not a cross-section. The cross-section is the face exposed.
* Let's stick to the most common geometric cross-sections for a triangular prism:
1. Parallel to base: Triangle
2. Perpendicular to base: Rectangle
3. Diagonal: Trapezoid (or sometimes Parallelogram)
* *Wait, looking at the incomplete image provided in the prompt's `Final Answer` slot requirement:* The user wants me to solve the problem. The problem is filling in the blanks for the prism row.
* Blank 1 (Parallel): Triangle
* Blank 2 (Perpendicular): Rectangle
* Blank 3 (Diagonal): Trapezoid
Let's double-check the "Diagonal" cut for a triangular prism. If you cut diagonally across the entire length, intersecting all 5 faces, it is a pentagon? No, usually these problems assume simple cuts.
- Cut parallel to base: Triangle.
- Cut perpendicular to base (vertical): Rectangle.
- Cut diagonally (slanted): This usually produces a Trapezoid.
Therefore, the shapes to fill in are:
1. Under "Cut parallel to base": Triangle
2. Under "Cut perpendicular to base": Rectangle
3. Under "Cut diagonally": Trapezoid
Final Answer:
The missing cross-sections for the prism are:
1. Cut parallel to base: Triangle
2. Cut perpendicular to base: Rectangle
3. Cut diagonally: Trapezoid
Parent Tip: Review the logic above to help your child master the concept of geometry cross section worksheet.
