Objects and Solids

• triangle

• trapezoid

• hexagon

• rectangle

• cone

• cylinder

• pyramid

• rectangular prism

When a plane intersects a square pyramid parallel to its base, the resulting cross section is a​ (a)   that ​ (b)   . Since the base of a square pyramid is a ​ (c)   , the cross section will also be a square. This occurs because the plane cuts through the pyramid at a level ​ (d)   to the base, maintaining the same proportions and angles. Therefore, the cross section retains the characteristics of a square, with four equal sides and four right angles. Understanding this concept helps in visualizing how three-dimensional shapes can be sliced to ​ (e)   figures.

• two-dimensional shape

• mirrors the shape of the base

• square

• parallel

• reveal two-dimensional

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

When a right cylinder is cut perpendicular to its base, the resulting cross section is a ​ (a)   . This is because the cut is made ​ (b)   to the circular base, slicing through the cylinder's ​ (c)   . The length of the rectangle is equal to the diameter of the cylinder's base, and the width is equal to the height of the cylinder. This geometric property is important in understanding how three-dimensional shapes can be ​ (d)   into two-dimensional shapes through cross sections.

• at a right angle

• rectangle

• height

• transformed

• circle

• cylinder

• rectangle

• triangular prism

When a right cylinder is cut parallel to its base, the resulting cross section is a ​ (a)   . This is because the ​ base of a right cylinder is a ​ (b)   , and slicing it parallel to this base means the cut is made at a constant height, maintaining the circular shape. The size of the circle depends on the distance from the base where the cut is made, but the ​ (c)   unchanged. This concept is important in geometry as it helps in understanding the properties of three-dimensional shapes and their​ (d)   .

• circle

• shape remains

• two-dimensional cross sections

• 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