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Geometry Help: Cross Sections Rotations Of 3d Objects

Review real example questions for Cross Sections Rotations Of 3d Objects in Geometry.

Question 1 / 10

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Which solid is formed by rotating a right triangle about one of its legs (that leg is the axis of rotation)?

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Question 1

Which solid is formed by rotating a right triangle about one of its legs (that leg is the axis of rotation)?

  1. A cone (correct answer)
  2. A cylinder
  3. A sphere
  4. A prism

Explanation: This problem examines solids of revolution created by rotating 2D shapes. The original shape is a right triangle with one 90-degree angle and two perpendicular legs. The axis of rotation is one of the legs (perpendicular sides), around which the triangle spins. As the right triangle rotates about one leg, the hypotenuse traces out a slanted circular surface while the other leg sweeps from the axis outward to create the base. The resulting solid is a cone with the rotating leg as its height and the other leg determining the base radius. Students might incorrectly choose cylinder, which comes from rotating rectangles, not triangles. To understand this, imagine spinning a right triangular flag on its pole—the flag creates a cone shape.

Question 2

A square pyramid is sliced by a plane that is parallel to its square base. Which shape results from the cross-section shown?

  1. Triangle
  2. Circle
  3. Square (correct answer)
  4. Rectangle

Explanation: This question tests understanding of cross-sections of pyramids. The solid is a square pyramid with a square base and triangular lateral faces meeting at an apex. The cutting plane is parallel to the square base, creating a horizontal slice at some height above the base. When a plane parallel to the base cuts through a pyramid, it intersects all four lateral faces, creating a shape similar to the base but smaller. Since the base is square and the pyramid tapers uniformly, the cross-section is also a square, just scaled down proportionally. Students might incorrectly choose triangle, thinking all pyramid cross-sections are triangular. To visualize, imagine slicing a pyramid-shaped block horizontally—each slice maintains the base's shape but gets smaller toward the apex.

Question 3

A right circular cylinder is sliced by a plane that is perpendicular to the bases and passes through the cylinder's central axis. Which description correctly identifies the resulting shape?

  1. A circle
  2. An ellipse
  3. A rectangle (correct answer)
  4. A triangle

Explanation: This question examines cross-sections of three-dimensional solids, specifically cylinders. The original solid is a right circular cylinder with circular bases and a curved lateral surface. The slicing plane is perpendicular to the bases and passes through the cylinder's central axis, cutting vertically through the entire height. When this vertical plane cuts through the cylinder, it intersects both circular bases along diameters and the curved surface along two straight lines parallel to the axis. The resulting cross-section is a rectangle with width equal to the cylinder's diameter and height equal to the cylinder's height. Students might incorrectly choose circle or ellipse, thinking all cylinder cross-sections are curved. To visualize this, imagine cutting a paper towel tube lengthwise down the middle—you see a rectangular shape.

Question 4

A right circular cylinder is shown standing upright. A slicing plane cuts the cylinder perpendicular to the circular bases and passes through the cylinder's central axis (so the plane is vertical and contains the axis).

Which shape results from the cross-section shown?

  1. Circle
  2. Rectangle (correct answer)
  3. Ellipse
  4. Trapezoid

Explanation: This question tests understanding of cross-sections and solids of revolution in geometry. The original solid is a right circular cylinder standing upright with circular bases. The slicing plane is vertical, perpendicular to the bases, and passes through the central axis. The plane intersects the cylinder along the height and through the diameter of the bases. This creates a rectangular cross-section with width equal to the diameter and height of the cylinder. A distractor is mistaking it for a circle, which happens with parallel slices. Imagine the slice step by step along the axis to confirm the straight-sided rectangle.

Question 5

A right circular cone is shown with its base on a horizontal plane. A slicing plane cuts the cone and passes through the apex, and the plane is also perpendicular to the base (so the plane contains the cone's axis).

Which shape results from the cross-section shown?

  1. Circle
  2. Isosceles triangle (correct answer)
  3. Rectangle
  4. Ellipse

Explanation: This question tests understanding of cross-sections and solids of revolution in geometry. The original solid is a right circular cone with a base on a horizontal plane. The slicing plane passes through the apex and is perpendicular to the base, containing the axis. The plane cuts along the height and through the base diameter. This produces an isosceles triangular cross-section with the base as the diameter and sides as generators. Confusing it with a circle ignores the axial cut through the apex. Picture the vertical slice step by step from apex to base to see the triangle emerge.

Question 6

A right circular cylinder is shown with a clearly marked slicing plane. The plane is parallel to the circular base and cuts the cylinder halfway between the top and bottom bases.

Which shape results from the cross-section shown?

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

Explanation: This problem involves finding cross-sections of three-dimensional solids, a key skill in geometry that also relates to solids of revolution. The original solid is a right circular cylinder, which has two parallel circular bases connected by a curved lateral surface. The slicing plane is parallel to the circular bases and intersects the cylinder midway between the top and bottom. When the plane slices parallel to the bases, it intersects the lateral surface in a uniform manner, creating a shape identical to the bases. This results in a circle because the cross-section mirrors the base's shape due to the cylinder's uniform radius. A common misconception is thinking it forms an ellipse, which might occur if the plane were angled, but here it's parallel, so it's a circle. To visualize, imagine slicing a can of soup horizontally step by step; each slice reveals a circular cross-section matching the can's ends.

Question 7

A plane intersects a regular tetrahedron such that it is parallel to one face and located at 14\frac{1}{4} the distance from that face to the opposite vertex. What is the relationship between the area of this cross-section and the area of the original face?

  1. The cross-section area is 14\frac{1}{4} of the original face area
  2. The cross-section area is 916\frac{9}{16} of the original face area (correct answer)
  3. The cross-section area is 34\frac{3}{4} of the original face area
  4. The cross-section area is 116\frac{1}{16} of the original face area

Explanation: In a regular tetrahedron, when a plane is parallel to a face at distance 14\frac{1}{4} from that face toward the opposite vertex, it's at distance 34\frac{3}{4} from the vertex. The linear scale factor is 34\frac{3}{4}, so the area scales as (34)2=916(\frac{3}{4})^2 = \frac{9}{16}. Choice A confuses linear and area scaling. Choice C uses the linear scale factor incorrectly. Choice D incorrectly squares 14\frac{1}{4}.

Question 8

A right circular cylinder has radius 44 cm and height 1010 cm. A plane slices the cylinder parallel to the circular bases (so the plane is horizontal). Which shape results from the cross-section shown?

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

Explanation: This problem involves finding the cross-section of a 3D solid. The original solid is a right circular cylinder with circular bases. The slicing plane is parallel to the circular bases, making it horizontal. As the plane cuts through the cylinder, it intersects the curved surface evenly at every point. The resulting cross-section is a circle identical in shape to the bases. A common misconception is assuming it forms a rectangle, which occurs with a perpendicular slice instead. To visualize, imagine slicing a tube parallel to its ends step by step to reveal the circular shape.

Question 9

A right circular cylinder is sliced by a plane that is perpendicular to the cylinder's circular bases and passes through the center axis of the cylinder. Which description correctly identifies the resulting shape?

  1. Circle
  2. Rectangle (correct answer)
  3. Ellipse
  4. Trapezoid

Explanation: This question examines cross-sections of 3D objects, specifically cylinders. The solid is a right circular cylinder with circular bases and a curved lateral surface. The cutting plane is perpendicular to the bases and passes through the cylinder's center axis, creating a vertical slice through the middle. This plane intersects the top circle along a diameter, the bottom circle along a diameter, and connects these with straight lines along the cylinder's height. The resulting cross-section is a rectangle with width equal to the cylinder's diameter and height equal to the cylinder's height. Students might incorrectly choose circle, confusing this with horizontal cuts. To visualize, imagine cutting a paper towel tube lengthwise through the center—you see a rectangular shape.

Question 10

A right circular cylinder is shown with a slicing plane that is perpendicular to the circular bases and passes through the cylinder's central axis.

Which description correctly identifies the resulting shape?

  1. A rectangle (correct answer)
  2. A circle
  3. A semicircle
  4. A trapezoid

Explanation: This problem involves finding cross-sections of three-dimensional solids, a key skill in geometry that also relates to solids of revolution. The original solid is a right circular cylinder, featuring circular bases and a rectangular height when unrolled. The slicing plane is perpendicular to the bases and passes through the central axis of the cylinder. As the plane cuts vertically through the center, it intersects the two bases along their diameters and the lateral surface along the height. This produces a rectangle, with the width equal to the diameter of the base and the height matching the cylinder's height. A distractor misconception is assuming a semicircle, perhaps confusing it with a non-central slice, but the central axis ensures a full rectangular shape. To transfer this, imagine unfolding the cylinder and tracing the plane's path step by step to see the rectangular outline emerge.