Describe Cross-Sections of 3D Figures - 7th Grade Math

1. A rectangular prism is sliced by a vertical plane perpendicular to the base, cutting from the top face straight down to the bottom face. What 2D shape is the cross-section?​
   Answer: Rectangle Tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: rectangular prism sliced horizontally (parallel to base) gives rectangle cross-section (cuts through vertical faces creating rectangular outline), sliced vertically gives rectangle (through opposite faces). Rectangular pyramid sliced horizontally gives smaller rectangle (parallel to base, similar shape decreasing toward apex), sliced vertically through apex gives triangle (apex is vertex, base edge is side, isosceles if through center). For example, rectangular prism sliced vertically showing rectangle cross-section (height and width forming straight sides), or rectangular pyramid vertical through apex showing triangle (three vertices: apex and two base corners), or cylinder horizontal showing circular cross-section (parallel to circular base). The correct cross-section identification for a rectangular prism sliced vertically perpendicular to the base is a rectangle, as the plane cuts straight through the height and across faces, producing right angles. A common error is thinking it's a circle (wrong, rectangle—no curves in prism), or an ellipse (wrong, vertical cut gives straight lines). Determining cross-section: (1) identify 3D figure (prism, pyramid, cylinder, cone), (2) identify slice orientation (horizontal=parallel to base, vertical=perpendicular to base, through specific features like apex/axis), (3) apply rules (parallel to base→same shape as base for prism, smaller for pyramid/cone; through apex→triangle; vertical through cylinder axis→rectangle), (4) name 2D shape (rectangle, triangle, circle, etc.). Key patterns: prism horizontal/vertical both rectangles (rectangular faces), pyramid horizontal rectangle but vertical triangle (apex creates point), cylinder/cone horizontal circles (circular bases), vertical through axis rectangles or triangles (cone apex).
2. A cylinder is sliced by a vertical plane that passes through the center axis of the cylinder. What 2D shape is the cross-section?
   Answer: Rectangle This question tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: cylinder horizontal gives circle (parallel to circular base maintains circular shape), vertical through axis gives rectangle. For example, a cylinder sliced vertically through the axis shows a rectangle cross-section (height as sides, diameter as width). The correct cross-section identification is a rectangle, as the plane cuts along the height and through the curved surface, unfolding to straight lines. A common error is claiming it's a circle (wrong, rectangle—vertical through axis, circle for horizontal). To determine the cross-section: (1) identify the 3D figure (cylinder), (2) identify slice orientation (vertical through axis), (3) apply rules (vertical through axis→rectangle), (4) name 2D shape (rectangle). Key patterns: cylinder horizontal circle but vertical rectangle, and common mistakes include confusing with ellipses (which occur for slanted slices).
3. Based on the rectangular prism shown, if plane P intersects the prism and creates a cross-section that includes parts of exactly 4 faces of the prism, which statement must be true about the resulting cross-section?
   Answer: The cross-section must be a parallelogram with at least one pair of parallel sides When a plane intersects exactly 4 faces of a rectangular prism, it creates a quadrilateral. Due to the parallel faces of the prism (opposite faces are parallel), the cross-section must be a parallelogram. The intersecting plane creates parallel edges where it cuts through parallel faces of the prism.
4. A rectangular prism is cut by a vertical plane that goes straight down from the top face to the bottom face (perpendicular to the base). What 2D shape is the cross-section?
   Answer: Rectangle This question tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: rectangular prism sliced horizontally (parallel to base) gives rectangle cross-section (cuts through vertical faces creating rectangular outline), sliced vertically gives rectangle (through opposite faces). For example, a rectangular prism sliced vertically perpendicular to the base shows a rectangle cross-section (sides matching the height and width of the faces it cuts). The correct cross-section identification is a rectangle, as the vertical slice cuts through the height and two opposite faces, forming a rectangular shape. A common error is claiming it's a circle (wrong, rectangle—prisms have rectangular faces, not circular). To determine the cross-section: (1) identify the 3D figure (rectangular prism), (2) identify slice orientation (vertical=perpendicular to base), (3) apply rules (vertical through faces→rectangle for prism), (4) name 2D shape (rectangle). Key patterns: prism horizontal/vertical both rectangles (rectangular faces), but common mistakes include confusing with pyramids where vertical slices give triangles.
5. A rectangular prism (like a cereal box) is sliced by a horizontal plane that is parallel to its rectangular base. What 2D shape is the cross-section?
   Answer: Rectangle This question tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: rectangular prism sliced horizontally (parallel to base) gives rectangle cross-section (cuts through vertical faces creating rectangular outline), sliced vertically gives rectangle (through opposite faces). For example, a rectangular prism sliced horizontally shows a rectangle cross-section (top and bottom edges parallel, sides straight—rectangular outline). The correct cross-section is a rectangle, as the horizontal slice parallel to the base of a rectangular prism produces a shape identical to the base. A common error is choosing triangle (wrong, as that's for vertical slices through a pyramid's apex, not a prism where sides are parallel and don't taper). To determine the cross-section: (1) identify the 3D figure as a rectangular prism, (2) note the slice is horizontal and parallel to the base, (3) apply the rule that parallel to the base in a prism gives the same shape as the base (rectangle), (4) name the 2D shape as rectangle. Key patterns include prisms yielding rectangles for both horizontal and vertical slices due to their uniform rectangular faces, unlike pyramids where horizontal slices are rectangles but vertical through the apex are triangles.
6. Which of these slices would produce a triangle shape?
   Answer: A vertical slice through the apex of a rectangular pyramid A vertical slice through the apex of a rectangular pyramid cuts from the single top point down through the base, forming a shape with three sides: a triangle. This matches choice D. Choice A gives a circular cross-section, since a horizontal slice of a cylinder parallel to its base always produces a circle matching the base. Choice B also gives a circle, since a horizontal slice of a cone parallel to its base produces a circular cross-section, not a point or triangle. Choice C gives a rectangle, since a horizontal slice of a rectangular prism parallel to its base is congruent to that rectangular base.
7. A model of a museum roof is shaped like a rectangular pyramid. A slice is made with a horizontal plane parallel to the base. What is the cross-section?
   Answer: A rectangle smaller than the base This question tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: rectangular pyramid sliced horizontally gives smaller rectangle (parallel to base, similar shape decreasing toward apex), sliced vertically through apex gives triangle (apex is vertex, base edge is side, isosceles if through center). For example, a rectangular pyramid sliced horizontally parallel to the base shows a smaller rectangle cross-section (similar to the base but scaled down toward the apex). The correct cross-section identification is a rectangle smaller than the base, as the pyramid tapers, making horizontal slices progressively smaller. A common error is thinking it's a triangle (wrong, rectangle—horizontal slices parallel to rectangular base, triangle only through apex). To determine the cross-section: (1) identify the 3D figure (rectangular pyramid), (2) identify slice orientation (horizontal=parallel to base), (3) apply rules (parallel to base→smaller similar shape for pyramid), (4) name 2D shape (rectangle). Key patterns: pyramid horizontal rectangle but vertical triangle (apex creates point), and common mistakes confuse with prisms where sizes don't shrink.
8. A cone is sliced by a horizontal plane parallel to its circular base. What 2D shape is the cross-section?
   Answer: Circle Tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: rectangular prism sliced horizontally (parallel to base) gives rectangle cross-section (cuts through vertical faces creating rectangular outline), sliced vertically gives rectangle (through opposite faces). Cone horizontal gives circle (smaller toward apex), vertical through apex gives isosceles triangle. For example, cone horizontal showing circular cross-section (parallel to base, even curve but smaller due to taper), or cone vertical through apex showing triangle (apex point and base diameter forming isosceles sides), or cylinder horizontal showing circular cross-section (parallel to circular base). The correct cross-section identification for a cone sliced horizontally parallel to the base is a circle, as the slice follows the curved surface evenly, producing a smaller version of the base. A common error is thinking it's a triangle (wrong, circle—parallel to circular base), or a rectangle (wrong, curved shape preserved). Determining cross-section: (1) identify 3D figure (prism, pyramid, cylinder, cone), (2) identify slice orientation (horizontal=parallel to base, vertical=perpendicular to base, through specific features like apex/axis), (3) apply rules (parallel to base→same shape as base for prism, smaller for pyramid/cone; through apex→triangle; vertical through cylinder axis→rectangle), (4) name 2D shape (rectangle, triangle, circle, etc.). Key patterns: prism horizontal/vertical both rectangles (rectangular faces), pyramid horizontal rectangle but vertical triangle (apex creates point), cylinder/cone horizontal circles (circular bases), vertical through axis rectangles or triangles (cone apex).
9. Consider the pyramid cross-section shown in the figure. A right rectangular pyramid is cut by a plane that is NOT parallel to the base and intersects all four triangular faces but does not pass through the apex or base. What is true about the resulting cross-section?
   Answer: It must be a quadrilateral but not necessarily similar to the base When a non-horizontal plane intersects all four triangular faces (but not apex or base), it creates a quadrilateral by intersecting exactly four faces. However, since the plane is not parallel to the base, the cross-section is not similar to the base rectangle. The angles and side ratios will be different from the original base.
10. A rectangular pyramid is sliced by a vertical plane that passes through the apex and the midpoint of one side of the base. What 2D shape is the cross-section?
    Answer: Triangle This question tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: rectangular pyramid sliced horizontally gives smaller rectangle (parallel to base, similar shape decreasing toward apex), sliced vertically through apex gives triangle (apex is vertex, base edge is side, isosceles if through center). For example, a rectangular pyramid sliced vertically through the apex and base midpoint shows a triangle cross-section (three vertices: apex and two points on the base edge). The correct cross-section is a triangle, as the vertical slice through the apex of a rectangular pyramid cuts to form three sides. A common error is choosing rectangle (wrong, as that's for horizontal slices; vertical through the apex includes the pointy top, creating a triangle, not a rectangle). To determine the cross-section: (1) identify the 3D figure as a rectangular pyramid, (2) note the slice is vertical, passing through the apex and base midpoint, (3) apply the rule that vertical through the apex gives a triangle, (4) name the 2D shape as triangle. Key patterns include pyramids yielding triangles for vertical apex slices due to the apex point, unlike prisms where vertical slices are rectangles, with mistakes often from not considering the apex's role in forming a vertex.
11. A rectangular pyramid is sliced by a vertical plane that passes through the apex (top point) and the center of the base. What 2D shape is the cross-section?
    Answer: Triangle This question tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: rectangular pyramid sliced horizontally gives smaller rectangle (parallel to base, similar shape decreasing toward apex), sliced vertically through apex gives triangle (apex is vertex, base edge is side, isosceles if through center). For example, a rectangular pyramid sliced vertically through the apex shows a triangle cross-section (three vertices: apex and two base corners). The correct cross-section identification is a triangle, as the plane passes through the apex point and cuts the base edge, forming three sides. A common error is claiming it's a rectangle (wrong, triangle—vertical through apex creates a pointed shape, not rectangular). To determine the cross-section: (1) identify the 3D figure (rectangular pyramid), (2) identify slice orientation (vertical through apex), (3) apply rules (through apex→triangle), (4) name 2D shape (triangle). Key patterns: pyramid horizontal rectangle but vertical triangle (apex creates point), and orientation matters—horizontal vs vertical differ significantly.
12. A cylinder (like a soup can) is sliced by a horizontal plane parallel to its circular base. What 2D shape is the cross-section?
    Answer: Circle This question tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: cylinder horizontal gives circle (parallel to circular base maintains circular shape), vertical through axis gives rectangle. For example, a cylinder sliced horizontally shows a circular cross-section (parallel to the circular base, maintaining the round shape). The correct cross-section is a circle, as the horizontal slice parallel to the base of a cylinder follows the base's shape. A common error is choosing rectangle (wrong, as that's for vertical slices through the axis; horizontal in cylinders yields circles, not rectangles). To determine the cross-section: (1) identify the 3D figure as a cylinder, (2) note the slice is horizontal and parallel to the base, (3) apply the rule that parallel to the base gives the same shape as the base (circle), (4) name the 2D shape as circle. Key patterns include cylinders and cones producing circles for horizontal slices due to their circular bases, unlike prisms with rectangular ones, with mistakes confusing orientation and claiming rectangles from horizontal cuts.
13. A cone (like an ice cream cone) is sliced by a vertical plane that passes through the apex and the center of the circular base. What 2D shape is the cross-section?
    Answer: Triangle Tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: rectangular prism sliced horizontally (parallel to base) gives rectangle cross-section (cuts through vertical faces creating rectangular outline), sliced vertically gives rectangle (through opposite faces). Cone horizontal gives circle (smaller toward apex), vertical through apex gives isosceles triangle. For example, cone vertical through apex showing triangle (apex point and base diameter forming isosceles sides), or cone horizontal showing circular cross-section (parallel to base but smaller), or rectangular pyramid vertical through apex showing triangle (three vertices: apex and two base corners). The correct cross-section identification for a cone sliced vertically through the apex and base center is a triangle, as the plane cuts from the point to the base edges, creating three sides. A common error is thinking it's a circle (wrong, triangle—through apex forms point and straight lines), or a rectangle (wrong, tapers to point). Determining cross-section: (1) identify 3D figure (prism, pyramid, cylinder, cone), (2) identify slice orientation (horizontal=parallel to base, vertical=perpendicular to base, through specific features like apex/axis), (3) apply rules (parallel to base→same shape as base for prism, smaller for pyramid/cone; through apex→triangle; vertical through cylinder axis→rectangle), (4) name 2D shape (rectangle, triangle, circle, etc.). Key patterns: prism horizontal/vertical both rectangles (rectangular faces), pyramid horizontal rectangle but vertical triangle (apex creates point), cylinder/cone horizontal circles (circular bases), vertical through axis rectangles or triangles (cone apex).
14. A rectangular pyramid (a pyramid with a rectangular base) is sliced by a horizontal plane parallel to its base. What 2D shape is the cross-section?​
    Answer: A smaller rectangle Tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: rectangular prism sliced horizontally (parallel to base) gives rectangle cross-section (cuts through vertical faces creating rectangular outline), sliced vertically gives rectangle (through opposite faces). Rectangular pyramid sliced horizontally gives smaller rectangle (parallel to base, similar shape decreasing toward apex), sliced vertically through apex gives triangle (apex is vertex, base edge is side, isosceles if through center). For example, rectangular pyramid sliced horizontally showing smaller rectangle cross-section (parallel to base but scaled down due to tapering sides), or rectangular pyramid vertical through apex showing triangle (three vertices: apex and two base corners), or cylinder horizontal showing circular cross-section (parallel to circular base). The correct cross-section identification for a rectangular pyramid sliced horizontally parallel to the base is a smaller rectangle, as the pyramid tapers, making the cross-section similar but reduced in size. A common error is claiming it's a rectangle the same size as the base (wrong, smaller unless at the base—pyramid sides slant inward), or thinking it's a triangle (wrong, rectangle parallel to rectangular base, triangle only through apex). Determining cross-section: (1) identify 3D figure (prism, pyramid, cylinder, cone), (2) identify slice orientation (horizontal=parallel to base, vertical=perpendicular to base, through specific features like apex/axis), (3) apply rules (parallel to base→same shape as base for prism, smaller for pyramid/cone; through apex→triangle; vertical through cylinder axis→rectangle), (4) name 2D shape (rectangle, triangle, circle, etc.). Key patterns: prism horizontal/vertical both rectangles (rectangular faces), pyramid horizontal rectangle but vertical triangle (apex creates point), cylinder/cone horizontal circles (circular bases), vertical through axis rectangles or triangles (cone apex).
15. A traffic cone is shaped like a cone. A slice is made with a horizontal plane parallel to the base. What 2D shape is the cross-section?
    Answer: Circle This question tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: cone horizontal gives circle (smaller toward apex), vertical through apex gives isosceles triangle. For example, a cone like a traffic cone sliced horizontally parallel to the base shows a circular cross-section (smaller than the base due to tapering). The correct cross-section identification is a circle, as the horizontal slice parallel to the circular base maintains the round shape but scales it down. A common error is thinking it's a triangle (wrong, circle—parallel to base, triangle for vertical through apex). To determine the cross-section: (1) identify the 3D figure (cone), (2) identify slice orientation (horizontal=parallel to base), (3) apply rules (parallel to base→smaller circle for cone), (4) name 2D shape (circle). Key patterns: cone horizontal circles but vertical triangles (apex creates point), and common mistakes confuse with prisms or non-circular shapes.
16. A soup can is shaped like a cylinder. You slice it with a horizontal plane parallel to the circular base. What 2D shape is the cross-section?
    Answer: Circle This question tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: cylinder horizontal gives circle (parallel to circular base maintains circular shape), vertical through axis gives rectangle. For example, a cylinder like a soup can sliced horizontally parallel to the base shows a circular cross-section (matching the base shape). The correct cross-section identification is a circle, as the horizontal slice parallel to the circular base preserves the round outline. A common error is thinking it's a rectangle (wrong, circle—parallel to circular base, rectangle for vertical slices). To determine the cross-section: (1) identify the 3D figure (cylinder), (2) identify slice orientation (horizontal=parallel to base), (3) apply rules (parallel to base→circle for cylinder), (4) name 2D shape (circle). Key patterns: cylinder/cone horizontal circles (circular bases), but vertical through axis rectangles or triangles (cone apex).
17. A rectangular prism is sliced by a vertical plane perpendicular to the base, cutting from the top face straight down to the bottom face. What 2D shape is the cross-section?
    Answer: Rectangle Tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: rectangular prism sliced horizontally (parallel to base) gives rectangle cross-section (cuts through vertical faces creating rectangular outline), sliced vertically gives rectangle (through opposite faces). Rectangular pyramid sliced horizontally gives smaller rectangle (parallel to base, similar shape decreasing toward apex), sliced vertically through apex gives triangle (apex is vertex, base edge is side, isosceles if through center). For example, rectangular prism sliced vertically showing rectangle cross-section (height and width forming straight sides), or rectangular pyramid vertical through apex showing triangle (three vertices: apex and two base corners), or cylinder horizontal showing circular cross-section (parallel to circular base). The correct cross-section identification for a rectangular prism sliced vertically perpendicular to the base is a rectangle, as the plane cuts straight through the height and across faces, producing right angles. A common error is thinking it's a circle (wrong, rectangle—no curves in prism), or an ellipse (wrong, vertical cut gives straight lines). Determining cross-section: (1) identify 3D figure (prism, pyramid, cylinder, cone), (2) identify slice orientation (horizontal=parallel to base, vertical=perpendicular to base, through specific features like apex/axis), (3) apply rules (parallel to base→same shape as base for prism, smaller for pyramid/cone; through apex→triangle; vertical through cylinder axis→rectangle), (4) name 2D shape (rectangle, triangle, circle, etc.). Key patterns: prism horizontal/vertical both rectangles (rectangular faces), pyramid horizontal rectangle but vertical triangle (apex creates point), cylinder/cone horizontal circles (circular bases), vertical through axis rectangles or triangles (cone apex).
18. A cone is sliced by a horizontal plane parallel to its circular base. What 2D shape is the cross-section?​
    Answer: Circle Tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: rectangular prism sliced horizontally (parallel to base) gives rectangle cross-section (cuts through vertical faces creating rectangular outline), sliced vertically gives rectangle (through opposite faces). Cone horizontal gives circle (smaller toward apex), vertical through apex gives isosceles triangle. For example, cone horizontal showing circular cross-section (parallel to base, even curve but smaller due to taper), or cone vertical through apex showing triangle (apex point and base diameter forming isosceles sides), or cylinder horizontal showing circular cross-section (parallel to circular base). The correct cross-section identification for a cone sliced horizontally parallel to the base is a circle, as the slice follows the curved surface evenly, producing a smaller version of the base. A common error is thinking it's a triangle (wrong, circle—parallel to circular base), or a rectangle (wrong, curved shape preserved). Determining cross-section: (1) identify 3D figure (prism, pyramid, cylinder, cone), (2) identify slice orientation (horizontal=parallel to base, vertical=perpendicular to base, through specific features like apex/axis), (3) apply rules (parallel to base→same shape as base for prism, smaller for pyramid/cone; through apex→triangle; vertical through cylinder axis→rectangle), (4) name 2D shape (rectangle, triangle, circle, etc.). Key patterns: prism horizontal/vertical both rectangles (rectangular faces), pyramid horizontal rectangle but vertical triangle (apex creates point), cylinder/cone horizontal circles (circular bases), vertical through axis rectangles or triangles (cone apex).
19. A cone is sliced by a vertical plane that passes through the apex and the center of the base. What 2D shape is the cross-section?
    Answer: Triangle This question tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: cone horizontal gives circle (smaller toward apex), vertical through apex gives isosceles triangle. For example, a cone sliced vertically through the apex and base center shows a triangular cross-section (apex point and base diameter forming sides). The correct cross-section is a triangle, as the vertical slice through the apex of a cone cuts to form three sides, often isosceles. A common error is choosing rectangle (wrong, as that's for cylinder vertical slices; cones taper to a point, creating a triangle). To determine the cross-section: (1) identify the 3D figure as a cone, (2) note the slice is vertical, passing through the apex and base center, (3) apply the rule that vertical through the apex gives a triangle, (4) name the 2D shape as triangle. Key patterns include cones yielding triangles for vertical apex slices due to the point, unlike cylinders' rectangles, with mistakes from not accounting for the apex's tapering effect.
20. Which slice would produce a triangle as the cross-section of a rectangular pyramid?
    Answer: A vertical slice that passes through the apex This question tests describing 2D cross-sections from slicing 3D figures: horizontal slices of prisms/pyramids (rectangles), vertical slices through apex (triangles), horizontal slices of cylinders/cones (circles), based on slice orientation. Cross-section shape depends on slice orientation and 3D figure: rectangular pyramid sliced horizontally gives smaller rectangle (parallel to base, similar shape decreasing toward apex), sliced vertically through apex gives triangle (apex is vertex, base edge is side, isosceles if through center). For example, a rectangular pyramid sliced vertically through the apex shows a triangle cross-section (three vertices: apex and two base points). The correct slice to produce a triangle is a vertical one passing through the apex, as it includes the point and base edge to form three sides. A common error is choosing horizontal parallel to the base (wrong, as that gives rectangles; triangles require the apex in the slice). To determine the cross-section: (1) identify the 3D figure as a rectangular pyramid, (2) evaluate each slice orientation, (3) apply rules like through apex for triangle versus parallel to base for rectangle, (4) select the vertical apex slice. Key patterns include pyramids needing apex inclusion for triangles, unlike prisms, with mistakes assuming horizontal slices taper to triangles instead of smaller rectangles.

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