Volume of Geometric Shapes

The following figure shows a container in the shape of a circular cylinder topped by a hemisphere. When the cylinder and the hemisphere both have radius of and the height of the cylinder is , what is the volume of this solid figure?

The cone-shaped water tank is to be filled with water at a constant speed as shown in the following figure. If it took minutes to get water up to a height of , how many more minutes would it take to completely fill the tank with water? (Here, the thickness of the water tank is ignored.)

Find the volume of one ball in the cylindrical container with a volume of when four balls of the same size fit perfectly into the container as shown in the following figure.

Find the volume of the frustum of the cone as shown in the following figure.

Find the volume of a frustum of a square pyramid as shown in the following figure.

Find the volume of a sphere that fits into a cube of volume , as shown in the following figure.

After filling a cylinder of the radius of the circle at the base and the height of with water, the sphere was immersed in it and taken out, and the level of the water became , as shown in the following figure. Find the surface area of the sphere in this case.

As shown in the following figure, there is a cylinder, a sphere that fits perfectly inside the cylinder, and a cone that also fits perfectly inside the cylinder. When the volume of the sphere is , what is the sum of the volumes of the cylinder and the cone?

What is the volume of a triangular pyramid cut from a rectangular parallelepiped to a plane passing through three vertices , , and ?

As shown in the following figure, a cube with an edge length of was cut through points , , and . When points , , and are mid-points of , , and , respectively, what is the volume of the remaining solid figure?

Find the volume of the solid obtained by rotating the shaded region shown below about the vertical line .

A hollow cylindrical metal pipe is long and its outer and inner diameters are and , respectively. Find the volume of the metal used in making the pipe.

A wooden pipe with a diameter of is drilled into a cylindrical hole of diameter of . When the volume of this wooden pipe after the hole has been drilled is , what is the surface area?

Find the value of when the amount of water in two rectangular parallelepiped containers is equal, as shown in the following figure. (Here, ignore the thickness of the container.)

Find the value of when the amount of water in two rectangular parallelepiped containers is equal, as shown in the following figure. (Here, ignore the thickness of the container.)

Find the volume of the solid figure that results from cutting a cube with an edge of into a plane passing through four points , , , and , as shown in the following figure. (Here, the two points and are the mid-points of and , respectively.)

The following figure is an regular octahedron that connects the intersection points of the diagonal of each side in a cube of volume . Find the volume of this regular octahedron.

The following figure shows the triangular pyramid , which is made by connecting the diagonal of a cube with an edge of . Find the volume of this triangular pyramid.

The following is a solid figure created by connecting midpoints of each edge of the face with the intersection of the diagonal of the face in a cube. When the length of an edge of the cube is , what is the volume of this figure?

Find the surface area of the pyramid that is produced in this case.

Find the volume of the pyramid that is produced.