A cube with edge length 6 is intersected by a plane that passes through one vertex and is perpendicular to the space diagonal through the opposite vertex. What is the shape and approximate area of the cross-section?
Answer: A regular hexagon with area approximately 273 square units determined by perpendicular intersection
When a plane passes through a vertex of a cube and is perpendicular to the space diagonal through the opposite vertex, it creates a regular hexagon. This hexagon passes through the midpoints of 6 edges of the cube. Each side of the hexagon has length 3√2 (half the face diagonal of the cube). The area of a regular hexagon with side length s is (3√3/2)s². With s = 3√2, the area is (3√3/2)(3√2)² = (3√3/2)(18) = 27√3. Choice A has the correct shape but wrong area calculation. Choice B has the wrong shape. Choice D has both wrong shape and area.
A right circular cylinder with radius 5 inches and height 12 inches is cut by two parallel planes. The first plane is perpendicular to the cylinder's axis and located 3 inches from the bottom. The second plane is also perpendicular to the axis and located 8 inches from the bottom. What is the shape of the solid region between these two cutting planes?
Answer: A cylinder with radius 5 inches and height 5 inches
When a cylinder is cut by planes perpendicular to its axis, each cutting plane creates a circular cross-section with the same radius as the original cylinder. The solid between the two planes is the portion of the original cylinder between heights 3 inches and 8 inches, which has height 8−3=5 inches. Since both cutting planes are perpendicular to the axis, they create parallel circular faces with radius 5 inches (same as the original cylinder). The resulting solid is therefore a cylinder with radius 5 inches and height 5 inches. Choice A is wrong because cylinders don't create frustums when cut perpendicularly. Choice C is wrong because cylinders don't have rectangular cross-sections. Choice D is wrong because perpendicular cuts of cylinders remain circular, not elliptical.
A regular hexagonal prism has a base with side length 4 cm and height 10 cm. A plane cuts the prism at an angle such that it intersects four of the six rectangular lateral faces and both hexagonal bases. What is the minimum number of sides the cross-sectional polygon can have?
Answer: 6 sides
The number of sides in a cross-section equals the number of faces the cutting plane intersects. A hexagonal prism has 8 faces total: 2 hexagonal bases and 6 rectangular lateral faces. The problem states the plane intersects 4 of the 6 lateral faces plus both hexagonal bases, giving 4+2=6 faces total. Therefore, the cross-section must have exactly 6 sides. The question asks for the minimum number, but since the plane must intersect both bases and exactly 4 lateral faces (as given), there's only one possibility: 6 sides. Choice A (4 sides) would only count the lateral faces. Choice B (5 sides) would mean intersecting 5 faces total. Choice D (8 sides) would require intersecting all 8 faces, but the problem specifies only 4 lateral faces are intersected.
A regular pentagonal prism is cut by a plane parallel to its pentagonal base but closer to one end than the other. The resulting cross-section will have which property?
Answer: The same shape and area as the original pentagonal base of the prism
When a plane cuts parallel to the base of any prism, the cross-section is congruent to the base regardless of where along the height the cut is made. This is because the sides of a prism are parallel to each other, so any plane parallel to the base will intersect all sides at the same relative position, creating an identical shape and size. Choice A is incorrect because it mentions only area, not shape. Choice B incorrectly suggests the shape changes. Choice D is impossible because if the area is smaller, the perimeter of a similar regular pentagon must also be smaller.
A regular tetrahedron is cut by a plane that passes through the midpoints of three edges that share a common vertex. What is true about the resulting cross-section?
Answer: It is an equilateral triangle with side length half that of the tetrahedron's edges
When a plane passes through the midpoints of three edges meeting at a vertex of a regular tetrahedron, it creates a cross-section that is an equilateral triangle. By the midpoint theorem, each side of this triangle connects midpoints of edges of the tetrahedron, making each side of the cross-section exactly half the length of the tetrahedron's edges. Choice B incorrectly states the triangle sides equal the original edge length. Choices C and D are incorrect because the symmetry of the regular tetrahedron ensures all three sides of the cross-section are equal.
A right circular cylinder is intersected by a plane that passes through the center of one circular base and is tangent to the opposite circular base. What curve describes the boundary of the cross-section?
Answer: An ellipse with major axis extending from center to tangent point distance
When a plane intersects a 3D object, the resulting cross-section depends on the angle and position of that plane. This question tests your understanding of conic sections formed by planar intersections of cylinders.
Picture this intersection: the plane passes through the center of one circular base and just touches (is tangent to) the opposite base at a single point. This creates an angled cut through the cylinder that's neither perpendicular nor parallel to the cylinder's axis. When you slice a cylinder at such an angle, you get an ellipse. The ellipse's major axis runs from the center point on one base to the tangent point on the opposite base, while its minor axis is constrained by the cylinder's circular cross-section. This makes choice D correct.
Let's examine why the other options fail: Choice A suggests a parabola, but parabolas form when planes intersect cones, not cylinders. Choice B describes a circle, which would only occur if the plane cut perpendicular to the cylinder's axis. Choice C mentions a hyperbola, but hyperbolas require the plane to intersect both branches of a double cone or pass through a cylinder in a way that creates two separate curves.
Remember this pattern: angled planar cuts through cylinders nearly always produce ellipses. The key insight is recognizing that the described intersection creates an elongated, oval-shaped boundary rather than any of the other conic sections. Focus on visualizing the geometric setup when tackling similar spatial reasoning problems.
A rectangular parallelepiped (box) with dimensions 4×6×8 is cut by a plane that passes through four vertices, no three of which lie on the same face. How many sides does the resulting cross-section polygon have?
Answer: 4 sides, creating a quadrilateral that spans multiple faces of the box
When a plane intersects a three-dimensional object, the resulting cross-section is determined by how many faces of the object the plane actually cuts through. For a rectangular parallelepiped (box), you need to visualize which faces the cutting plane encounters.
Since the plane passes through four vertices with no three on the same face, these four vertices must form a skew quadrilateral that spans across the box. When you connect any four vertices of a rectangular box under this constraint, the plane necessarily intersects exactly four faces of the box. Each intersection with a face creates one edge of the cross-sectional polygon, so the result is a quadrilateral with four sides.
Choice A is incorrect because three vertices would create a triangular cross-section, but the problem specifically states four vertices are involved. Choice B assumes the plane intersects all six faces, which is geometrically impossible when passing through only four vertices with the given constraints. Choice C suggests five intersections, but this would require the plane to pass through more vertices or have a different orientation than what's described.
The key insight is that the number of sides in the cross-section equals the number of faces the cutting plane intersects. With four vertices positioned as described, the plane can only intersect four faces, creating a four-sided polygon.
Study tip: For any plane-solid intersection problem, count the faces that the plane actually cuts through—this directly gives you the number of sides in the resulting cross-section polygon.
A cube is sliced by a plane that passes through three vertices that are not on the same face. What is the shape of the resulting cross-section?
Answer: An equilateral triangle with sides longer than the cube's edge length
When a plane passes through three vertices of a cube that are not on the same face, it creates an equilateral triangle. However, the sides of this triangle are face diagonals of the cube, not edges. If the cube has edge length s, each side of the triangle has length s√2, which is longer than the cube's edge length. Choice A incorrectly assumes the triangle sides equal the cube's edges. Choice C would occur if the plane passed through vertices creating unequal sides. Choice D is incorrect because the symmetry of the cube ensures all three sides are equal.
A rectangular pyramid with a square base is sliced by a plane parallel to one of its triangular faces. What shape is the cross-section?
Answer: A quadrilateral with two parallel sides of different lengths and two non-parallel sides
When a plane is parallel to a triangular face of a square pyramid, it intersects the opposite triangular face and two adjacent triangular faces, creating a quadrilateral. This quadrilateral is a trapezoid because one pair of sides (where the plane intersects opposite faces) are parallel, while the other pair (from adjacent faces) are not parallel. Choice A would only be correct if the plane coincided with the triangular face itself. Choice C incorrectly assumes the plane intersects all faces. Choice D incorrectly suggests a rectangle, which would require two pairs of parallel sides.
A cube with edge length 8 units is intersected by a plane that passes through three vertices: one vertex of the top face and two adjacent vertices of the bottom face. What is the shape of the cross-section?
Answer: An isosceles triangle with two sides of length 8√2 units and one side of length 8 units
Let's place the cube in a coordinate system with vertices from (0,0,0) to (8,8,8). If we choose the top vertex at (0,0,8) and two adjacent bottom vertices at (0,0,0) and (8,0,0), we can find the triangle's side lengths. The distance from (0,0,0) to (8,0,0) is 8 units (along the bottom edge). The distance from (0,0,8) to (0,0,0) is 8 units (vertical edge). The distance from (0,0,8) to (8,0,0) is √(8² + 0² + 8²) = √128 = 8√2 units (diagonal). Therefore, we have an isosceles triangle with two sides of length 8√2 units (the diagonals from the top vertex to each bottom vertex) and one side of length 8 units (the bottom edge).
A right circular cylinder with radius 6 and height 10 is cut by a plane that makes a 30° angle with the base. If the plane passes through the entire cylinder, what type of curve forms the boundary of the cross-section?
Answer: An ellipse with major axis length determined by the cylinder's height and angle
When a plane cuts through a cylinder at an angle (not perpendicular to the axis), the cross-section is always an ellipse. The major axis length depends on the angle of the cut and the cylinder's dimensions. At 30°, the plane intersects the cylinder obliquely, stretching the circular cross-section into an elliptical shape. Choice A incorrectly assumes a circular cross-section, which only occurs when the plane is perpendicular to the cylinder's axis. Choices C and D describe conic sections that result from cutting cones, not cylinders.
A solid sphere of radius 10 cm contains a cylindrical hole drilled completely through its center. The cylinder has radius 4 cm and its axis passes through the center of the sphere. If a plane cuts through this object perpendicular to the cylinder's axis at a distance 6 cm from the sphere's center, what is the shape of the cross-section?
Answer: A circle with a circular hole, where the outer circle has radius 8 cm and the hole has radius 4 cm
The cross-section will be an annulus (ring shape) formed by the intersection of the plane with both the sphere and the cylindrical hole. For the outer boundary: the plane cuts the sphere at distance 6 cm from center, so the radius of the circular cross-section is 102−62=100−36=64=8 cm. For the inner boundary: since the cutting plane is perpendicular to the cylinder's axis and the cylinder has radius 4 cm throughout its length, the hole appears as a circle with radius 4 cm. Therefore, the cross-section is a circle with radius 8 cm containing a circular hole of radius 4 cm. Choice B incorrectly calculates the outer radius as 6 cm. Choice C uses the full sphere radius instead of the cross-section radius. Choice D ignores the cylindrical hole entirely.
A right circular cone with base radius 8 and height 12 is sliced by a plane parallel to its base at a height of 4 units from the apex. What is the radius of the circular cross-section?
Answer: 38 units, calculated by proportional reduction from the original base radius
In a cone, cross-sections parallel to the base create similar circles. The radius of the cross-section is proportional to the distance from the apex. Using similar triangles: radius/distance from apex = base radius/total height, so r/4 = 8/12 = 2/3, giving r = 4 × 2/3 = 8/3. Choice B incorrectly assumes the radius equals the height from apex. Choice C incorrectly subtracts values that shouldn't be subtracted. Choice D inverts the proportion and uses incorrect arithmetic.
A regular octahedron (8 triangular faces) is sliced by a plane parallel to two opposite faces. If the plane is positioned exactly halfway between these faces, what shape is the cross-section?
Answer: A square with sides parallel to four of the octahedron's edges at the widest point
A regular octahedron can be visualized as two square pyramids joined at their bases. When sliced by a plane parallel to two opposite triangular faces and positioned at the center, the plane intersects the octahedron at its widest point, which is a square. This square has its vertices at the midpoints of four edges of the octahedron and represents the 'equator' of the octahedron. Choice A incorrectly suggests a hexagon. Choice C is wrong because the cross-section is not triangular. Choice D incorrectly assumes the cross-section reflects all eight faces.
