Rotation to Solids

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

• 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

When a rectangle is continuously rotated around one of its sides, it forms a ​ (a)   known as a ​ (b)   . Imagine the rectangle standing upright, with one of its sides fixed as an axis. As the rectangle spins around this axis, it sweeps out a cylindrical shape. The height of the cylinder is equal to the length of the side around which the rectangle rotates, and the radius of the cylinder is equal to half the length of the opposite side. This process is similar to how a potter shapes clay on a ​ (c)   to create a vase.

• three-dimensional shape

• cylinder

• spinning wheel

• rectangular prism

• cylinder

• sphere

• cone

• cone

• pyramid

• prism

• sphere

When a circle is continuously rotated about its diameter, it forms a three-dimensional object known as a ​ (a)   . Imagine spinning a flat circle around a straight line that ​ (b)   As it spins, every point on the circle traces out a path in space, creating a perfectly round shape. This shape is called a sphere, which is like a ​ (c)   . A sphere has a surface where every point is the same distance from the center. This distance is called the radius. Spheres are common in everyday life, like ​ (d)   and globes.

• basketballs

• sphere

• goes through its center.

• ball

• cone

• prism

• sphere

• cylinder