Naming and Slicing Solids

When a square pyramid is intersected by a plane passing through its vertex and ​ (a)   the resulting cross section is a ​ (b)   . This is because the plane slices through the apex of the pyramid and extends down to the base, cutting through two opposite edges of the square base. The ​ (c)   formed has its vertex at the ​ (d)   and its base along the line where the plane intersects the square base. The sides of the triangle are formed by the slant heights of the pyramid, making it an isosceles triangle if the pyramid is ​ (e)   .

• perpendicular to its base,

• triangle

• apex of the pyramid

• regular

• square

• triangle

• pentagon

• rectangle

• circle

• cylinder

• rectangle

• triangular prism

• cone

• circle

• triangle

• rectangle

A regular pyramid is a three-dimensional shape with a polygonal base and triangular faces that meet at a common point called the apex. When you take a cross section of a regular pyramid that includes its altitude, the resulting shape is a ​ (a)   . This triangle is formed by slicing the pyramid from the apex ​ (b)   , creating a ​ (c)   plane. The ​ (d)   of the pyramid is the perpendicular line from the apex to the center of the base, and it becomes one of the sides of this triangular cross section. This cross section helps in understanding the pyramid's symmetry and dimensions.

• triangle

• down to the midpoint of the base

• altitude

• vertical

When a plane intersects a hexagonal prism perpendicularly to its base, the cross section formed is a ​ (a)   . To understand why, consider the structure of a hexagonal prism. It has two hexagonal ​ (b)   and rectangular lateral faces connecting corresponding sides of the hexagons. When the plane cuts through the prism ​ (c)   , it slices through these lateral faces. Since the plane is perpendicular, it intersects each lateral face along a straight line, forming a ​ (d)   . The height of this rectangle is the same as the height of the prism, and its width is equal to the distance between two opposite sides of the ​ (e)   .

• rectangle

• bases

• perpendicularly to the base

• hexagon

• triangle

• trapezoid

• hexagon

• rectangle

When a plane intersects a sphere, the shape formed by this cross section is a ​ (a)   . To understand why, imagine slicing through a round ball with a ​ (b)   . The edge where the sheet cuts through the ball creates a perfectly round shape. This is because a sphere is symmetrical ​ (c)   , and any flat cut through it will always result in a circle. The size of the circle depends on how far the plane is from the ​ (d)   . If the plane passes through the center, the circle will be the ​ (e)   possible, known as a great circle.

• circle

• flat sheet

• in all directions

• center of the sphere

• largest

When a two-dimensional cross section of a three-dimensional object is taken, the shape of the cross section depends on the object and the ​ (a)   If the cross section is a triangle, the three-dimensional object cannot be a ​ (b)   . A sphere, when sliced, always results in a ​ (c)   , regardless of the angle or position of the cut. Other objects, like a cube or a pyramid, can produce ​ (d)   when sliced at certain angles. Therefore, a sphere or cylinder are the only objects that ​ (e)   have a triangular cross section.

• angle of the cut.

• sphere

• circular cross section

• triangular cross sections

• cannot

• cone

• cylinder

• pyramid

• rectangular prism