When a plane enters and exits through opposite faces at a 45° angle, the resulting cross-section depends on which pair of opposite faces it intersects and the exact entry/exit points. The plane could create either a rectangle (if cutting straight across) or a parallelogram (if cutting at an angle across different dimensions), with varying areas depending on the intersection path.

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.

Cross-sections parallel to the base are similar squares. At 1/3 height from base (2/3 from apex), linear scale factor is 2/3. At 2/3 height from base (1/3 from apex), linear scale factor is 1/3. Since the lower section has area 36 and scale factor 2/3, the base area is 36 ÷ (2/3)² = 81. The upper section area is 81 × (1/3)² = 9 square units.

In a right rectangular pyramid, horizontal cross-sections are similar to the base with a scaling factor based on distance from the apex. At 3/4 up from base means 1/4 down from apex. The side length scales proportionally: remaining fraction × original side = 1/4 × 10 = 2.5 units.

When a plane passes through the midpoints of three edges meeting at a vertex of a cube, it intersects exactly three faces, creating a triangle. Since all edges of a cube are equal and the cutting points are all midpoints, the resulting triangle is equilateral with all sides equal and all angles 60°.

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.

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).

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.

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 like a cereal box sliced horizontally parallel to the base shows a rectangle cross-section (top and bottom edges parallel, sides straight—rectangular outline). The correct cross-section identification is a rectangle, as the horizontal slice parallel to the base mirrors the rectangular base shape. A common error is thinking a horizontal prism slice gives a triangle (wrong, it's a rectangle—prism faces are rectangles, horizontal cuts parallel giving rectangle). To determine the cross-section: (1) identify the 3D figure (rectangular prism), (2) identify slice orientation (horizontal=parallel to base), (3) apply rules (parallel to base→same shape as base for prism), (4) name 2D shape (rectangle). Key patterns: prism horizontal/vertical both rectangles (rectangular faces), and orientation matters—horizontal vs vertical can differ in other shapes like pyramids.

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 vertical through apex showing triangle (three vertices: apex and two base corners), or rectangular prism sliced horizontally showing rectangle cross-section (top and bottom edges parallel, sides straight—rectangular outline), or cylinder horizontal showing circular cross-section (parallel to circular base). The correct identification for producing a triangle is a vertical slice through the apex of a rectangular pyramid, as it cuts from the point to the base, forming three sides. A common error is choosing horizontal slice of a prism (wrong, rectangle—not tapering to point), or horizontal cone (wrong, circle—parallel to base). 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).