Middle School Math Quiz: Describe Cross Sections Of 3d Figures
20 questions · exam conditions
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Describe Cross Sections Of 3d FiguresQuestion 1 of 20

A right rectangular pyramid has a square base. Two different planes each cut the pyramid parallel to the base: one at 1/3 the height from the base, another at 2/3 the height from the base. If the lower cross-section has an area of 36 square units, what is the area of the upper cross-section?

16 square units, because the area decreases by the height ratio
24 square units, because the area decreases linearly with height
9 square units, because the area scales with the square of linear dimensions
12 square units, because the area decreases by half the height difference
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Middle School Math Quiz

Middle School Math Quiz: Describe Cross Sections Of 3d Figures

Practice Describe Cross Sections Of 3d Figures in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Describe Cross Sections Of 3d Figures, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A right rectangular pyramid has a square base. Two different planes each cut the pyramid parallel to the base: one at 1/3 the height from the base, another at 2/3 the height from the base. If the lower cross-section has an area of 36 square units, what is the area of the upper cross-section?

  1. 16 square units, because the area decreases by the height ratio
  2. 24 square units, because the area decreases linearly with height
  3. 9 square units, because the area scales with the square of linear dimensions (correct answer)
  4. 12 square units, because the area decreases by half the height difference
Explanation: 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.

Question 2

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?

  1. Triangle (correct answer)
  2. Parallelogram
  3. Rectangle
  4. Circle
Explanation: 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.

Question 3

Which slice would produce a triangle as the cross-section?

Choose the best option.

  1. A horizontal slice parallel to the base of a cylinder
  2. A vertical slice through the apex of a rectangular pyramid (correct answer)
  3. A horizontal slice parallel to the base of a rectangular prism
  4. A horizontal slice parallel to the base of a cone
Explanation: 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).

Question 4

A rectangular prism (like a cereal box) is sliced by a horizontal plane that is parallel to the base. What 2D shape is the cross-section?

  1. Rectangle (correct answer)
  2. Triangle
  3. Circle
  4. Trapezoid
Explanation: 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 horizontally showing rectangle cross-section (top and bottom edges parallel, sides straight—rectangular outline), 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 horizontally parallel to the base is a rectangle, as the slice cuts evenly through the vertical faces, producing a shape identical to the base. A common error is thinking a horizontal prism slice gives a triangle (wrong, rectangle—prism faces are rectangles, horizontal cuts parallel giving rectangle), or confusing it with a pyramid slice. 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).

Question 5

A right rectangular pyramid has a square base with side length 10 units and height 12 units. If a horizontal slice is made at a height that is 3/4 of the way up from the base to the apex, what is the side length of the resulting square cross-section?

  1. 2.5 units, because the pyramid shrinks uniformly toward the apex (correct answer)
  2. 7.5 units, because three-fourths of the original length remains
  3. 4.0 units, because the height ratio creates proportional scaling
  4. 6.25 units, because the area decreases by the height ratio
Explanation: 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.

Question 6

A rectangular prism is sliced by a vertical plane that is perpendicular to the base and cuts from the front face straight to the back face (like making a straight up-and-down cut). What 2D shape is the cross-section?

  1. Rectangle (correct answer)
  2. Triangle
  3. Ellipse
  4. Circle
Explanation: 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 from front to back shows a rectangle cross-section (height and length forming the outline). The correct cross-section is a rectangle, as the vertical slice perpendicular to the base in a rectangular prism produces a rectangular shape. A common error is choosing circle (wrong, as prisms have no curved surfaces; circles come from cylinders or cones). To determine the cross-section: (1) identify the 3D figure as a rectangular prism, (2) note the slice is vertical, perpendicular to the base and straight through, (3) apply the rule that vertical slices in prisms give rectangles, (4) name the 2D shape as rectangle. Key patterns include prisms always giving rectangles for horizontal or vertical slices due to straight faces, with mistakes claiming impossible shapes like circles from non-cylindrical figures.

Question 7

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?​

  1. A circle
  2. A triangle
  3. A rectangle the same size as the base
  4. A smaller rectangle (correct answer)
Explanation: 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).

Question 8

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?

  1. Circle
  2. Rectangle (correct answer)
  3. Triangle
  4. Ellipse
Explanation: 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).

Question 9

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?

  1. A circle
  2. A smaller rectangle (correct answer)
  3. A triangle
  4. A rectangle the same size as the base
Explanation: 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).

Question 10

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?

  1. It must be a quadrilateral similar to the base rectangle
  2. It must be a quadrilateral but not necessarily similar to the base (correct answer)
  3. It could be either a triangle, quadrilateral, or pentagon shape
  4. It must be a rectangle with sides parallel to the base edges
Explanation: 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.

Question 11

Which of these slices would produce a triangle shape?

  1. A horizontal slice parallel to the base of a cylinder
  2. A horizontal slice parallel to the base of a cone
  3. A horizontal slice parallel to the base of a rectangular prism
  4. A vertical slice through the apex of a rectangular pyramid (correct answer)
Explanation: 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.

Question 12

A cube is sliced by a plane that passes through the midpoints of exactly three edges that all meet at the same vertex. How many sides does the resulting cross-section have, and what type of polygon is it?

  1. 3 sides, forming an equilateral triangle with all angles equal (correct answer)
  2. 4 sides, forming a quadrilateral with two pairs of equal sides
  3. 3 sides, forming an isosceles triangle with two equal sides only
  4. 6 sides, forming a hexagon with alternating side lengths
Explanation: 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°.

Question 13

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?

  1. The cross-section must be a rectangle with sides parallel to the prism edges
  2. The cross-section must be a quadrilateral but may not have any parallel sides
  3. The cross-section must be a parallelogram with at least one pair of parallel sides (correct answer)
  4. The cross-section could be either a triangle or quadrilateral depending on the angle
Explanation: 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.

Question 14

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?

  1. Triangle
  2. Pentagon
  3. Circle
  4. Rectangle (correct answer)
Explanation: 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.

Question 15

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?

  1. Circle
  2. Rectangle
  3. Triangle (correct answer)
  4. Parallelogram
Explanation: 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.

Question 16

A rectangular pyramid is sliced by a plane that is parallel to the base. Which statement best describes the cross-section?

  1. It is a rectangle similar to the base, but smaller. (correct answer)
  2. It is a rectangle the same size as the base.
  3. It is a circle because the slice is flat.
  4. It is a triangle because pyramids have triangular faces.
Explanation: 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 parallel to the base shows a rectangle cross-section similar to the base but smaller (due to the tapering sides). The correct statement is that it is a rectangle similar to the base, but smaller, as horizontal slices produce scaled-down versions of the base shape. A common error is saying it's a triangle because pyramids have triangular faces (wrong, rectangle—parallel slice matches base shape, not faces). To determine the cross-section: (1) identify the 3D figure (rectangular pyramid), (2) identify slice orientation (parallel to base), (3) apply rules (parallel to base→smaller similar rectangle), (4) name 2D shape (rectangle). Key patterns: pyramid tapers so horizontal slices shrink, unlike prisms; common mistakes include assuming same size or confusing with vertical slices.

Question 17

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?

  1. Triangle (correct answer)
  2. Square
  3. Circle
  4. Rectangle
Explanation: When a cone is sliced by a vertical plane that passes through both the apex and the center of the circular base, the cut goes from the single point at the top down to a straight line across the base, creating a shape with three straight sides, which is a triangle, matching choice A. Choice C, a circle, would only result from a horizontal slice parallel to the base, not a vertical slice through the apex. Choice D, a rectangle, would only happen with shapes that have parallel flat sides all the way up, like a cylinder or prism, which a cone does not have. Choice B, a square, does not match any standard cross-section of a cone at all. The key idea is that any vertical slice through a cone's apex tapers to a single point, which always produces a triangular cross-section.

Question 18

A cone is sliced by a vertical plane that passes through the apex (the tip) and the center of the base. What 2D shape is the cross-section?

  1. Rectangle
  2. Triangle (correct answer)
  3. Circle
  4. Trapezoid
Explanation: 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 shows a triangle cross-section (apex as vertex, base diameter as base side). The correct cross-section identification is a triangle, as the plane passes through the tip and cuts the base, forming three sides. A common error is claiming it's a circle (wrong, triangle—vertical through apex, circle for horizontal). To determine the cross-section: (1) identify the 3D figure (cone), (2) identify slice orientation (vertical through apex), (3) apply rules (through apex→triangle), (4) name 2D shape (triangle). Key patterns: cone horizontal circle but vertical triangle (apex creates point), and orientation matters—horizontal vs vertical differ like in pyramids.

Question 19

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?

  1. Triangle
  2. Rectangle
  3. Circle (correct answer)
  4. Trapezoid
Explanation: 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).

Question 20

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?

  1. Circle
  2. Trapezoid
  3. Triangle (correct answer)
  4. Rectangle
Explanation: 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.