Unit 5 Lesson 1 Solids of Rotation

When a rectangle is continuously rotated about one of its sides, a ​ ​ (a)   solid is formed. In this case, if the rectangle is rotated about side w‍ , the resulting solid is a ​ (b)   . The side w‍  becomes the​ (c)   , and the opposite side of the rectangle forms the circular bases of the cylinder. The height of the cylinder is equal to the length of ​ (d)    , while the radius of the circular bases is equal to half the length of the rectangle's other side. This geometric transformation illustrates how ​ (e)   shapes can create three-dimensional figures through rotation.

• cylinder
• axis of rotation
• side w
• two-dimensional
• three-dimensional

• Pyramid

• rectangular prism

• cone

• cylinder

When a rectangle is continuously rotated about one of its sides, labeled as side S‍ , it generates a three-dimensional shape known as a ​ (a)   . The side S‍  becomes the ​ (b)   , and the opposite side remains parallel to it, forming the ​ (c)   . The height of the cylinder is equal to the length of ​ (d)    , while the radius of the circular base is half the length of the rectangle's other side. This process illustrates the concept of rotational symmetry and helps in understanding how two-dimensional shapes can create ​ (e)   -dimensional objects through rotation.

• cylinder
• axis of rotation
• circular bases of the cylinder
• side S
• three

• rectangular prism

• square pyramid

• cylinder

• cone

When a right triangle, such as triangle RST, is rotated around one of its legs, a ​ (a)   -dimensional object is formed. In this case, rotating triangle RST around leg RS creates a solid known as a right circular ​ (b)   . The leg RS acts as the ​ (c)   , and the hypotenuse RT sweeps out the curved surface of the cone. The point R becomes the ​ (d)   , while the base of the cone is a circle formed by the rotation of the other leg, ST. This geometric transformation highlights the relationship between two-dimensional shapes and their three-dimensional counterparts.

• three
• cone
• axis of rotation
• apex of the cone

• a pyramid with a square base

• an isosceles triangle

• a right triangle

• a cone

When triangle ABC, representing a metal flag, spins rapidly around pole AD, it creates the ​ (a)   . This phenomenon occurs due to the rotational motion, which causes the triangle to sweep out a volume in space. The shape formed is a ​ (b)   , with the pole AD acting as ​ (c)   . The base of the cone is the path traced by point C, while the apex is at point A. This transformation from a two-dimensional triangle to a three-dimensional cone illustrates ​ (d)   , such as rotational symmetry and the conservation of angular momentum.

• the axis of rotation
• illusion of a three-dimensional shape
• cone
• principles of geometry and physics

• sphere

• pyramid

• right circular cylinder

• cone

When a quarter of a circle, centered at the origin, is rotated about the y-axis, it generates a three-dimensional shape known as a "​ (a)   ." The quarter circle is the part of the circle in the first quadrant. As it rotates around the y-axis, each point on the quarter circle traces out a ​ (b)   , forming a hemisphere. This shape is ​ (c)   , with the flat side lying on the xy-plane and the curved surface extending outward.

• hemisphere
• circular path
• half of a sphere

• Cone

• Sphere

• Cylinder

• Hemisphere