Cross-Sections, Rotations of 3D Objects - Geometry
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What is the 2D cross-section of a cylinder cut by a plane through its axis (perpendicular to bases)?
What is the 2D cross-section of a cylinder cut by a plane through its axis (perpendicular to bases)?
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A rectangle. Axial cuts through cylinder height create rectangular cross-sections.
A rectangle. Axial cuts through cylinder height create rectangular cross-sections.
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What is the 2D cross-section of a cylinder cut by a plane parallel to its bases?
What is the 2D cross-section of a cylinder cut by a plane parallel to its bases?
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A circle. Parallel cuts to circular bases always produce circles.
A circle. Parallel cuts to circular bases always produce circles.
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What is the 2D cross-section of a cone cut by a plane parallel to its base?
What is the 2D cross-section of a cone cut by a plane parallel to its base?
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A circle. Parallel cuts to cone base maintain circular shape, scaled down.
A circle. Parallel cuts to cone base maintain circular shape, scaled down.
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What is the 2D cross-section of a cone cut by a plane through its vertex and axis?
What is the 2D cross-section of a cone cut by a plane through its vertex and axis?
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An isosceles triangle. Cuts through vertex and axis create symmetric triangular sections.
An isosceles triangle. Cuts through vertex and axis create symmetric triangular sections.
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What is the 2D cross-section of a sphere cut by any plane?
What is the 2D cross-section of a sphere cut by any plane?
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A circle. All plane cuts of a sphere create circular cross-sections.
A circle. All plane cuts of a sphere create circular cross-sections.
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What is the 2D cross-section of a cube cut by a plane parallel to a face?
What is the 2D cross-section of a cube cut by a plane parallel to a face?
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A square. Parallel cuts to cube faces preserve square shape.
A square. Parallel cuts to cube faces preserve square shape.
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What is the 2D cross-section of a rectangular prism cut by a plane parallel to a face?
What is the 2D cross-section of a rectangular prism cut by a plane parallel to a face?
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A rectangle. Parallel cuts to rectangular faces maintain rectangular shape.
A rectangle. Parallel cuts to rectangular faces maintain rectangular shape.
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What is the cross-section of a triangular prism cut by a plane parallel to its triangular bases?
What is the cross-section of a triangular prism cut by a plane parallel to its triangular bases?
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A triangle congruent to the base. Parallel cuts to triangular bases preserve triangular shape.
A triangle congruent to the base. Parallel cuts to triangular bases preserve triangular shape.
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What is the cross-section of any prism cut by a plane parallel to its bases?
What is the cross-section of any prism cut by a plane parallel to its bases?
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A congruent copy of the base. Parallel cuts to any prism base create congruent shapes.
A congruent copy of the base. Parallel cuts to any prism base create congruent shapes.
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What is the cross-section of any pyramid cut by a plane parallel to its base?
What is the cross-section of any pyramid cut by a plane parallel to its base?
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A similar copy of the base. Parallel cuts to pyramid base create similar, scaled shapes.
A similar copy of the base. Parallel cuts to pyramid base create similar, scaled shapes.
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What is the cross-section of a square pyramid cut parallel to its base?
What is the cross-section of a square pyramid cut parallel to its base?
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A square. Parallel cuts to square base create smaller squares.
A square. Parallel cuts to square base create smaller squares.
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What is the cross-section of a triangular pyramid cut parallel to its base?
What is the cross-section of a triangular pyramid cut parallel to its base?
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A triangle. Parallel cuts to triangular base create smaller triangles.
A triangle. Parallel cuts to triangular base create smaller triangles.
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What is the cross-section of a cube cut by a plane passing through 4 vertices (a space diagonal cut)?
What is the cross-section of a cube cut by a plane passing through 4 vertices (a space diagonal cut)?
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A rectangle. Space diagonal cuts through opposite vertices form rectangles.
A rectangle. Space diagonal cuts through opposite vertices form rectangles.
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What is the cross-section of a cube cut by a plane passing through 3 vertices not all on one face?
What is the cross-section of a cube cut by a plane passing through 3 vertices not all on one face?
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A triangle. Three non-coplanar vertices define a triangular cross-section.
A triangle. Three non-coplanar vertices define a triangular cross-section.
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What is the cross-section of a cube cut by a plane that produces a regular hexagon?
What is the cross-section of a cube cut by a plane that produces a regular hexagon?
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A regular hexagon. Specific angled cuts through cube can create hexagonal sections.
A regular hexagon. Specific angled cuts through cube can create hexagonal sections.
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Which solid is generated by rotating a right triangle about one of its legs?
Which solid is generated by rotating a right triangle about one of its legs?
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A right circular cone. Right triangle rotated about leg creates circular cone.
A right circular cone. Right triangle rotated about leg creates circular cone.
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Which solid is generated by rotating a semicircle about its diameter?
Which solid is generated by rotating a semicircle about its diameter?
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A sphere. Semicircle rotated about diameter creates complete sphere.
A sphere. Semicircle rotated about diameter creates complete sphere.
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Which solid is generated by rotating a circle about its diameter (in its plane)?
Which solid is generated by rotating a circle about its diameter (in its plane)?
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A sphere. Circle rotated about diameter creates sphere volume.
A sphere. Circle rotated about diameter creates sphere volume.
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Which solid is generated by rotating an isosceles trapezoid about its line of symmetry?
Which solid is generated by rotating an isosceles trapezoid about its line of symmetry?
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A frustum of a cone. Trapezoid rotated about symmetry axis creates truncated cone.
A frustum of a cone. Trapezoid rotated about symmetry axis creates truncated cone.
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Which solid is generated by rotating a square about a line through its center and parallel to a side?
Which solid is generated by rotating a square about a line through its center and parallel to a side?
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A right circular cylinder. Square rotated about central line creates cylinder.
A right circular cylinder. Square rotated about central line creates cylinder.
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Which solid is generated by rotating a rectangle about a line in its plane that does not intersect it?
Which solid is generated by rotating a rectangle about a line in its plane that does not intersect it?
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A hollow cylinder (cylindrical shell). Non-intersecting rotation creates hollow cylindrical shape.
A hollow cylinder (cylindrical shell). Non-intersecting rotation creates hollow cylindrical shape.
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Which solid is generated by rotating a line segment about a line that intersects it at one endpoint?
Which solid is generated by rotating a line segment about a line that intersects it at one endpoint?
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A right circular cone (degenerate if length $=0$). Endpoint intersection creates cone from rotating segment.
A right circular cone (degenerate if length $=0$). Endpoint intersection creates cone from rotating segment.
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Which solid is generated by rotating a line segment about a line through its midpoint and perpendicular to it?
Which solid is generated by rotating a line segment about a line through its midpoint and perpendicular to it?
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A disk (a circle filled in). Perpendicular rotation about midpoint creates filled disk.
A disk (a circle filled in). Perpendicular rotation about midpoint creates filled disk.
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Identify the solid formed by rotating a right triangle about its hypotenuse.
Identify the solid formed by rotating a right triangle about its hypotenuse.
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A double cone (two cones sharing a base). Rotation about hypotenuse creates two cones base-to-base.
A double cone (two cones sharing a base). Rotation about hypotenuse creates two cones base-to-base.
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What is the cross-section of a cone cut by a plane not through the vertex and not parallel to the base?
What is the cross-section of a cone cut by a plane not through the vertex and not parallel to the base?
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An ellipse. Non-parallel, non-vertex cuts create elliptical conic sections.
An ellipse. Non-parallel, non-vertex cuts create elliptical conic sections.
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What conic section is formed when a plane cuts a cone parallel to a generating line?
What conic section is formed when a plane cuts a cone parallel to a generating line?
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A parabola. Cuts parallel to generating lines create parabolic sections.
A parabola. Cuts parallel to generating lines create parabolic sections.
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What conic section is formed when a plane cuts both nappes of a double cone?
What conic section is formed when a plane cuts both nappes of a double cone?
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A hyperbola. Cuts through both cone parts create hyperbolic sections.
A hyperbola. Cuts through both cone parts create hyperbolic sections.
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Identify the 2D cross-section of a square pyramid cut by a plane through the apex and perpendicular to the base.
Identify the 2D cross-section of a square pyramid cut by a plane through the apex and perpendicular to the base.
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A triangle. Vertical cuts through apex create triangular sections.
A triangle. Vertical cuts through apex create triangular sections.
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Identify the 2D cross-section of a rectangular prism cut by a plane through 3 edges meeting at a vertex.
Identify the 2D cross-section of a rectangular prism cut by a plane through 3 edges meeting at a vertex.
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A triangle. Three meeting edges define triangular cross-section plane.
A triangle. Three meeting edges define triangular cross-section plane.
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Which cross-section results when a plane cuts a prism at an angle to the bases (not parallel)?
Which cross-section results when a plane cuts a prism at an angle to the bases (not parallel)?
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A parallelogram (often a rectangle in right prisms). Angled cuts to prism bases create parallelogram shapes.
A parallelogram (often a rectangle in right prisms). Angled cuts to prism bases create parallelogram shapes.
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