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