Cavalieri’s Principle states that if two solids of equal height are cut into cross-sections by parallel planes, and the corresponding cross-sections on each plane always have equal areas, then the two solids have the same volume.

A cross-section is the two-dimensional figure formed by intersecting a solid with a plane.

These diagrams show that the shape of the cross-section depends on how the plane intersects the solid.

The cube root of a number \(x\), written \(\sqrt[3]{x}\), is the number \(y\) whose cube is \(x\).

That is, \(y^3 = x\). So, \(\sqrt[3]{x}=y\).

Example: \(2^3 = 8\). So, \(\sqrt[3]{8}=2\).

Density is a measure of how tightly the amount of matter in a substance is packed into the space it takes up. That is, density is the mass of a substance per unit volume.

\(\text{density}=\frac{\text{mass}}{\text{volume}}\)

Any flat surface on a three-dimensional figure is a face.

• A cube has 6 faces that are all squares.
• A cylinder has 2 faces that are both circles.

An oblique solid is not exactly upright—it seems to lean over at an angle.

• Prisms and cylinders are said to be oblique if the directed line segment that defines the translation between the bases is not perpendicular to the bases.
• A cone is said to be oblique if a line drawn from its apex at a right angle to the plane of its base does not intersect the center of the base. The same definition applies to pyramids whose bases are figures with a center point, such as a square or a regular pentagon.

A prism is a three-dimensional figure composed of two parallel, congruent faces (called bases) connected by parallelograms. A prism is named for the shape of its bases. For example, if a prism’s bases are pentagons, it is called a pentagonal prism.

A pyramid is a three-dimensional figure that has one special face called the base. All of the other faces are triangles that meet at a single vertex called the apex. A pyramid is named for the shape of its base. For example, if a pyramid’s base is a hexagon, it is called a hexagonal pyramid.

A right solid is exactly upright—it does not seem to lean over at an angle.

• Prisms or cylinders are said to be right if the directed line segment that defines the translation between the bases is perpendicular to the bases.
• A cone is said to be right if a line drawn from its apex at a right angle to the plane of its base passes through the center of the base. The same definition applies to pyramids whose bases are figures with a center point, such as a square or a regular pentagon.

A solid of rotation is a three-dimensional figure formed by rotating a two-dimensional figure around a line called the axis of rotation.

In this diagram, the axis of rotation is the dashed line. The green triangle is rotated around the axis of rotation to form a solid of rotation.

A sphere is the set of all points in a three-dimensional space that are the same distance from a center point. All the cross-sections of a sphere are circles.