Solids with Known Cross Section Areas

A cross section of a solid is mathematically defined as the plane surface that results from the intersection of a plane with the solid. Therefore, a cross section is an infinitesimally thin slice of the solid. Solids commonly encountered in solid geometry include the sphere, ellipsoid, cylinder, cube, cone, and pyramid. Because solids are 3-dimensional figures, they have volumes that can be calculated. A cross section of a solid is a 2-dimensional shape, for which an area (called the cross-sectional area of the solid) can be calculated.

Although a cross section of a solid is mathematically a "slice" of the solid, it is easier to visualize a cross section as one of the cut ends produced when the solid is cut completely in two. For example, imagine that the earth were cut in two along the equator. Each cut surface (cross section) of the earth would be flat and in the shape of a circle. The area of a circle is determined from r², where r is the radius of the circle. Therefore, the cross-sectional area of the earth could be calculated using this formula.

Imagine that the earth is perfectly round, like a sphere. Then, a cross section produced by cutting from the North Pole to the South Pole would be identical in area to the cross section obtained from the cut across the equator. So would any complete cut that goes through the center of the earth. This is because of the symmetry of a sphere at all points around its center. Unless otherwise specified, a cross section is generally considered to be a section including the center of the solid (i.e., the intersection of its axis).

An ellipsoid can be visualized as a sphere stretched along one of its axis. Imagine an earth stretched way out of proportion and shaped like a bullet, with the North Pole at one end of the bullet, and the South Pole at the other end. If this Earth is cut along the equator, the resulting cross section is a circle. However, if this Earth is cut from the North Pole to the South Pole, the resulting cross section is an ellipse. The formula for the area of an ellipse is r₁r₂, where r₁ and r₂ are the radii of the ellipse along its short and long axis, respectively.

A cylinder also has two different cross sections, depending on which axis is cut along. A cut through the curved surfaces (usually the short axis) provides a circular cross section. A cut from one flat end to the other flat end (usually the long axis) provides a rectangular cross section. The area of a rectangle is calculated from its width times its length.

A cube cut along the x, y or z axis has a square cross section. No matter which of these axis the cube is cut along, the resulting cross sections will be squares with the same area. The area of a square is calculated from its width times its length (which are equal to each other).

All of the solids described so far have been cut through their centers to provide a cross section. For the cone and the pyramid, cross sections are sometimes better defined as being parallel or perpendicular to their bases. For example, if a cone is cut across its curved surfaces along any plane parallel to its base, a circular cross section is obtained. This is known as one of the conic sections. A cross section taken near the base will have a larger area than one taken near the point of the cone. If the cone is cut from its point to its base (through the center of the base), a triangular cross section is obtained. The area of a triangle is calculated from ½ times its base times its height.

Consider a pyramid with a square for its base and four triangular sides meeting at a point at the top (the apex). If this pyramid is cut across a plane parallel to its base, a square cross section will be obtained. Just as for the cone, a cross section taken near the base will have a larger area than one taken near the apex. If this pyramid is cut from its apex to its base (through the center of the base), the resulting cross section is triangular.

Remember that a cross section is defined by the intersection of a plane with a solid. The orientation of that plane to the solid must be defined in order to obtain the correct cross section and calculate the cross-sectional area.